#### NUMBER SYSTEM CONCEPT

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** NUMBER SYSTEM **

**Types of Numbers: Part-1**

In this first article of our topic ‘Numbers’, we deal with ‘Numbers’ themselves. What are numbers? Where do we see and meet them? Are they of different types? Can we learn these distinct types of numbers?

Well, numbers are all around us, floating all over the place. They demand our attention and careful evaluation. A number here or there can make a huge difference. In fact a misplaced zero or decimal is the difference between one being poor or rich.

Such being their importance, it is critical we begin with the most basic concepts in Mathematics: types of numbers. But before we do that, let’s define what is a number line:

**Number line**:A number line is line where all the numbers are allocated their positions. The origin of the number line starts from zero and it continues to infinity, on either side.

**Few Basic Types of Numbers**

**Positive Numbers** : Numbers which are to the right of zero are said to be positive numbers.

For example 1, 3 ,1.2 , 2.6 , 7 etc.

**Negative Numbers** : Numbers which are to the left of zero are said to be negative numbers.

For example -1, -5 , -7.2 , -2.5 , -9 etc.

**Counting Numbers**:Counting numbers are those numbers which are well managed on the number line with the difference of 1. The smallest counting number on the number line is 1.

**Natural Numbers:** Numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11…and so on are called natural numbers. They are also called positive integers. Also we can say that the other name for counting numbers is natural numbers.The lowest natural number is 1.

**Whole Numbers:**Whole numbers are numbers without fractions, they take integral values. Whole numbers are those numbers which start from 0 or we can say if 0 is included in set of counting numbers (natural numbers) then we get set of whole numbers.Remember whole numbers would always take non- negative integral values

**Types of Numbers: Part-2**

**Integers: **It is combination of both positive and negative numbers lying on the number line including zero. Remember zero is an integer. The integers are the natural numbers, their negatives, and the number zero like …,-3,-2,-1,0,1,2,3,… and they go on forever in both directions of the number line.

**Rational numbers: **Rational numbers are those, which can be written in the form of a ratio of x/ y, where the denominator y is non-zero.

**Irrational numbers**:** **Numbers, which are non-terminating and non- recurring (non-repeating) decimals are said to be irrational or we can say irrational numbers are those, which are not rational, that is those numbers that cannot be written in the form of a ratio x/y. For example = 1.414213….. , √3 =1.732050…..

**Fractions: **Fractions are those numbers, which are in the form of p/q where q is non-zero. For example 4/5, 5/6, 6/7 etc.

**Real Numbers**: Any number which can be plotted on the number line is a real number .The number can be positive or negative in nature. For example it may be like as 3, 4, 5, 6, -6, -5, -4, -3, -2…..

**Prime Numbers and Composite numbers**

In the list of natural numbers, there are two types of natural numbers.

One type is called the PRIME numbers and the other is called the COMPOSITE numbers.

(1 is not included in this list i.e.1 is neither prime nor composite)

**Prime numbers: **The numbers, which have exactly two factors, 1 and the number itself, are called prime numbers.

For example 2, 3, 5, 7, 11, 13, 31, 43,………………or we can say that the numbers, which are not divisible, by any number except one and itself, are called prime numbers**. **There are 25 prime numbers between 1 and 100.

** ****Composite number: **The composite numbers are those numbers which can be written as the product of prime numbers in a unique way.**Or** we can say that the number which is the product of two or more than two distinct or same prime numbers is said to be a composite number.

Example 4, 6, 8,…

Note that composite number always has more than two factors and prime number cannot be a composite number.

**Rounding numbers:**When the number is approximated to the nearest possible integral value to maintain the accuracy of the data then the number is called rounded.

**Properties of Numbers**

The following is a handy list of tips that you can remember about numbers (think about each one of these):

- The number line goes on till infinity in both directions, which is indicated by the arrows.
- The numbers on the number line are indicated by their respective signs, which shows that the line includes both positive and negative numbers.
- The integer zero is neutral and neither positive or negative.
- The number in the halfway of 1 and 2 is 1.5 and in the half way of -1 and -2 is -1.5.

- Between two rational numbers we can always find a rational number.
- Numbers, which are terminating and non-recurring are rational numbers. Similarly numbers, which are non-terminating and recurring are rational numbers.
- Set of natural numbers is contained in set of integers which is contained in set ofrational numbers which is further contained in set of real numbers, which is further contained in set of complex numbers.

- Addition as well as product of two real numbers is a real number.
- Two real numbers can be added or multiplied in either order i.e. Addition and multiplication of real numbers is commutative.
- Two real numbers cannot be subtracted or divided in either order i.e. Subtraction and Division of real numbers is not commutative.
- In set of real numbers we don’t define square root of negative numbers.

Rather we define set of complex numbers for this purpose. Any complex number z ={ a + ib , where a and b belongs to set of real numbers and i= }

**Properties of Zero**

- a x 0 =0 always for any real number a.
- a +0 = a always for any real number a.
- a – 0 = a always and 0 – a = -a for any real number a.
- 0/a =0 when “a” is a non-zero real number.
- a/0 is not defined i.e. we don’t define division by zero.
- a
^{0}=1for any non-zero real number a. - 0
^{0}is not defined.

**Prime numbers **

Numbers, which don’t have any factor other than 1 or itself, are said to be prime numbers.

For example 2, 3, 5, 7, 19, 29, 31………………

Or we can say that the numbers which are not divisible by any other number except 1 and itself, are called prime numbers.

There are 25 prime numbers between 1 and 100. In general we say there are infinite prime numbers.** ****Prime Numbers and Composite numbers **

In the list of natural numbers there are two types of natural numbers

One type is called PRIME numbers and other one is called COMPOSITE numbers.

**Properties of Prime Numbers**

- If
*p*is any prime number and p divides a product*p**divides ‘m’ or p divides ‘n’ or both.* - Number of prime numbers is uncountable. i.e. there are infinite prime numbers.
- Every Prime number has exactly two factors or divisors.

For example: 13 is prime number as the divisors of 13 are 1 and 13. - There is only one even prime i.e. 2 and all other primes are odd.
- G.C.D of prime numbers is always 1. The numbers whose G.C.D is 1 are said to be co-prime thus we say two primes are always co-prime.
- Two numbers are co-prime if their HCF is 1 .For example G.C.D. (21, 25)=1 and hence 21 and 25

are co- prime. - The Twin Primes are pair of primes of the form (p, p+2).
- It is conjectured that there are an infinite number of twin primes.
- The term odd prime refers to any prime number greater than 2. For example: 3, 5,7, …………..
- First Prime Number is 2.
- 1 is neither prime nor composite number.
- All prime numbers greater than 3 can be expressed in the form of 6n-1 or 6n+1 i.e.

all prime numbers,which are greater than 3 leave remainder 1 or 5 when divided by 6.

*List of prime numbers with number ranges:*

**Prime Numbers and Composite Numbers**

**How to check whether a number is prime or not?**

To check whether a given number N is prime or not, first find the square root of that number N and then approximate that to immediately lower integer (say n) and write down all the prime numbers less than that integer (n). Then check the divisibility of the given number N by all the prime numbers we have written in previous step, if it is not divisible by any of the prime numbers then given number N is prime.

Let us write algorithm for the same**Step 1**: Find square root of N, call it as K (Just find approximate values)**Step 2**: Write down all the prime numbers less than K.**Step 3**: Check divisibility of N with these prime numbers, which we have got in Step 2.**Step 4**: If N is not divisible by any of the prime numbers then N is prime.

**Example:**

Let us check whether 211 is prime or not?**Solution:****Step 1**: We find square root of 211 i.e. K=√211 = 14.52**Step 2**: We write all primes less than 14.52 i.e. 2, 3, 5, 7, 11 and 13.**Step 3**:Since 211 is not divisible by any of these prime numbers, hence 211 is a prime number.** **

**Example:**

Let us check whether 313 is prime or not?**Solution:****Step 1**: We find square root of 313 i.e. K=√311 = 17.69**Step 2**: We write all primes less than 17.69 i.e. 2, 3, 5, 7, 11, 13 and 17**Step 3**:Since 313 is not divisible by any of these prime numbers, hence 313 is a prime number.

**Composite numbers**

Number which is the product of two or more than two distinct or same prime numbers is said to be composite number **Or** we say that if a number has more than two factors then it is said to be a composite number.

For example 4, 6, 8, 15………..are all composite numbers

We can write 4 = 2 × 2,

6 = 2 × 3.*Some points to remember*

- 1 is neither prime nor composite.
- 2 is the only prime number which is even. Rest all prime numbers are odd.

**Properties of Addition of Natural Numbers:**Sum of 1

^{st ‘}n’ Natural numbers = 1 + 2 + 3 + …. + n = n(n+1)/2

Sum of squares of 1^{st }‘n’ natural number = 1^{2} + 2^{2} + 3^{2 }+ ….. + n^{2} = n(n+1)(2n+1)/6

Sum of cubes of 1^{st }‘n’ Natural number = 1^{3 }+ 2^{3} + 3^{3} + ….. + n^{3} = [n(n+1)/2]^2

Sum of 1^{st} ‘n’ Odd natural number = 1+ 3 + 5 + ….. + (2n – 1) = n^{2}

Sum of 1st ‘n’ even natural numbers = 2 + 4 + 6+ …….2n = n (n+1)

Sum of squares of 1^{st }‘n’ Odd natural number = 1^{2 }+ 3^{2} + 5^{2} + ….. + (2n – 1) ^{2} = n(2n−1)(2n+1)/3

Sum of squares of 1^{st} ‘n’ Even natural number = 2^{2} + 4^{2} + 6^{2} + ….. + (2n)^{2} = 2n(n+1)(2n+1)/3

Sum of cubes of 1^{st } ‘n’ Odd natural number = 1^{3 }+ 3^{3 }+ 5^{3} + ….. + (2n -1)^{3} = n^{2} (2n^{2} -1)

Sum of cubes of 1^{st } ‘n’ Even natural number = 2^{3} + 4^{3} + 6^{3} + ….. + (2n)^{3 }= 2 [n (n + 1)]^{2}

**Some examples for the above properties:**

*Find the sum of:*

*(A) First 30 natural numbers*

*(B) Squares of first 30 natural numbers*

*(C) Cubes of first 30 natural numbers*

*(D) First 30 odd natural numbers*

*(E) Squares of first odd 30 natural numbers*

*(F) Squares of first even 30 natural numbers*

*(G) Cubes of first odd 30 natural numbers*

*(H) Cubes of first even 30 natural numbers*

**Tricks with numbers**

**Number of times a particular digit (except 0) appears in a certain range:**

The concept, the title of the post talks about, is best understood with the help of examples. This is our way of practical learning and using problem types to understand concepts.**Example 1: Calculate how many times digit 3 appears in first hundred natural numbers?****Solution:** Well, there are two series of three’s here:**1**. Those set of numbers, which have 3 at unit place. 3, 13,……..,83 ,93, Here we have Ten 3s at the unit’s place**2**. Those set of numbers, which have 3 at ten`s place. 30,31…39.Here we have ten 3s at the ten’s place. So a total of 20 threes make an appearance here.

**Example 2: Calculate how many times digit 3 appears in natural numbers, which are less than one thousand?**

Solution: In previous example we calculated number of 3 s present till 100 are 20.

Now we look at three digit numbers and calculate how many times digit 3 appears in numbers between 100 and 1000 Well, there are three types of series here:**1.** Those set of numbers, which have **3 at unit place.**

103, 113, 123, ……..,193

203, 213 ,………………., 293

………………………………….

903,913,………………….993

Here we have 90 numbers having 3 at the unit’s place.**2.** Those set of numbers, which have **3 at ten`s place.**

130, 131, 132, ……..,139

230, 231, 232, ……..,239

………………………………….

930,931,………………….939

Here we have 90 numbers having 3 at the ten’s place

**3.** Those set of numbers, which have **3 at hundred`s place**.

301, 301, 302, ……..,399

Here we have hundred 3s at the hundred’s place.

Thus total number of times digit 3 appears in natural numbers, which are less than one thousand are 20+90+90+100=300

**Key Observations:***From 1-9, the digit 3 is used 1 time**From 1 – 99, the digit 3 is used 20 times.**From 1 – 999, the digit 3 is used 300 times**From 1 – 9999, the digit 3 is used 4000 times and so on*

Note: Above is true for any digit except 0.

**Unit digit of Perfect Squares**

**Properties regarding Unit Digit of Perfect Square**

Squares: There is a definite relationship between the unit digits when square of a number is considered, we will see that one by one.

– If unit digit of a perfect square is 1 then ten’s digit has to be Even

E.g. 81, 121, 441, 961 all are Perfect Square having unit digit 1 and tens digit is even.

– If unit digit of a number is 2 then it can’t be a perfect square.

– If unit digit of a number is 3 then it can’t be a perfect square.

– If unit digit of a perfect square is 4 then ten’s digit has to be Even.

E.g. 64, 144, 484 all are Perfect Square having unit digit 4 and tens digit is even.

– If unit digit of a perfect square is 5 then ten’s digit has to be 2

E.g. 25, 225, 625,1225 all are Perfect Square having unit digit 5 and tens digit is 2.

– If unit digit of a perfect square is 6 then ten’s digit has to be Odd. e.g. 256, 576, 676, 1296 etc.

– If unit digit of a number is 7 & 8 then it can’t be a perfect square

– If unit digit of a perfect square is 9 then ten’s digit has to be Even.

E.g. 49, 169, 529 all are Perfect Square having unit digit 9 and tens digit is even.

– If unit digit of a perfect square is 0 then ten’s digit has to be 0.

E.g. 100,400,900 all are Perfect Square having unit digit 0 and tens digit is 0.

– If a number ends with 2, 3, 7 or 8 then it can’t be perfect square.

-Square of any natural number has last two digits same as that of last two digits of squares of first twenty–five natural numbers. For example if we calculate square of

88 we get 7744. You can notice that last two digits are 44, which are same as last two digits of the square of 12 i.e. 144.

**Magic of Numbers ****Magical 1**

3 x 37 = 111

33 x 3367 = 111,111

333 x 333667 = 111,111,111

3333 x 33336667= 111,111,111,111

33333 x 3333366667= 111,111,111,111,111

333333 x 333333666667 = 111,111,111,111,111,111

Consider the L.H.S of the above numbers. The number of 3s in both the numbers are same and 6 is one less than the number of 3 and there is only one 7. Then in the final product, N would appear three times the number of threes in the first number on the LHS.

**6 when multiplied with 7 give 4 and 2 **

6 x 7 = 42

66 x67 = 4422

666 x 667 = 444222

6666 x 6667 = 44442222**For multiplication such as the above:**

The number of digits in both the numbers is same, the 2^{nd} number has one 6 less than the number of 6’s the 1^{st} number has .

The number of 4’s and 2’s is equal to the number of 6’s in the 1^{st} number

**Cyclic numbers **

142857 is called as the cyclic number, since its digits are rotated around when multiplied by any number from 1 to 6

142857 x 1 = 142857

142857 x 5= 714285

142857 x 4= 571428

142857 x 6 = 857142

142857 x 2= 285714

142867 x 3 = 428571

The same method helps us calculating fractions involving 7 in denominator

We know 1/7 = 0.142857

Thus 2/7 will be a value close to 0.28 and directly we will write exact answer as numbers repeat in cyclic order as shown in diagram. Hence exact answer will be 0.285714

Similarly 3/7 will be a value close to 0.42 and directly we will write exact answer as numbers repeat in cyclic orderas shown in diagram. Hence exact answer will be 0.428571

Similarly 4/7 will be a value close to 0.56 and directly we will write exact answer as numbers repeat in cyclic orderas shown in diagram. Hence exact answer will be 0.571428

Similarly 5/7 will be a value close to 0.7 and directly we will write exact answer as numbers repeat in cyclic orderas shown in diagram. Hence exact answer will be 0.714285

Similarly 6/7 will be a value close to 0.84 and directly we will write exact answer as numbers repeat in cyclic orderas shown in diagram. Hence exact answer will be 0.857142

**Properties of Consecutive Integers**

We have studied about integers, integers are combination of both positive and negative numbers lying on the number line including zero. There are few important properties regarding integers that you should know:

- Zero is also an Integer.
- Integers which follow one another are called consecutive integers.
- If we multiply a number by an integer and resultant value remains same then that integer is 1.

**Consecutive Integers: **The integers which follow one another are called consecutive integers.

For example 3, 4, 5, 6 are consecutive integers. An individual random number can never be

a consecutive integer.

**Note: **If we have consecutive integers in a set M from a to b i.e. M ={a,a+1,a+2,……………………..,b}.

Then number of elements in the set M is b–a+1.

**Example: **Set M consists of natural numbers from 75 to 199. Find the numbers of elements in set M ?

**Solution**: Question would have been very easy if we are asked number of natural numbers from 1 to 199. The answer to this is obviously 199. So we can also conclude it same way. From 75 to 199 we are not counting first 74 natural numbers. Hence the answer should be 199 -74= 125**Some more points about Consecutive Numbers ****TOOLTIP 1**

If there is odd number of digits in the set of consecutive numbers like set of three consecutive numbers (4,5,6,) or 5 digits are there, say (3,4,5,6,7), then in this case the sum of all integers is always divisible by the number of digits present in the set.

For example 2+3+4 = 9 is divisible by number of digits i.e. 3.

For example 1+2+3+4+5 = 15 is divisible by 5.

**TOOLTIP 2**

On the other hand ,If there are even number of digits in set of consecutive numbers like set of four consecutive numbers (4,5,6,7) or 6 digits are there, say (3,4,5,6,7,8), then in this case the sum of all integers is never divisible by the number of digits present in the set .

For example 2+3+4+5 = 14 is not divisible by number of digits i.e. 4

For example 1+2+3+4+5+6 = 21 is not divisible by 6

**Order of Operation **

While solving questions involving mathematical operations involving addition, subtraction, multiplication and division we follow a particular order while performing a calculation. To perform calculations we follow BODMAS.

BODMAS stands for

B = Bracket

O = of

D = Division

M= Multiplication

A = Addition

S= Subtraction**Let us see examples based on this****Example 1**:

What is the value of N= {(4 +5)of4}/3 + 2 ?**Solution:**

We will solve bracket first

{(9) x 4}/3 + 2

= 12 + 2

= 14

**Example 2**:

What is the value of N= {(12÷4× 2 +5) of4}/2−1 ?Solution:We will solve terms inside bracket first={(12÷4× 2 +5) of4}/2−1={(3 ×2 +5) of4}/2−1 (We divided 12 and 4 first)={(6 +5) of4}/2−1 (We multiplied 3 and 2)={(11) of4}/2−1 (We added 6 and 5)={44}/2−1 (We wrote 11 of 4)= 22−1(We divided 44 and 2)=21(We subtracted 1 from 22)">What is the value of N= {(12÷4× 2 +5) of4}/2−1 ?Solution:We will solve terms inside bracket first={(12÷4× 2 +5) of4}/2−1={(3 ×2 +5) of4}/2−1 (We divided 12 and 4 first)={(6 +5) of4}/2−1 (We multiplied 3 and 2)={(11) of4}/2−1 (We added 6 and 5)={44}/2−1 (We wrote 11 of 4)= 22−1(We divided 44 and 2)=21(We subtracted 1 from 22)What is the value of N= {(12÷4× 2 +5) of4}/2−1 ?Solution:We will solve terms inside bracket first={(12÷4× 2 +5) of4}/2−1={(3 ×2 +5) of4}/2−1 (We divided 12 and 4 first)={(6 +5) of4}/2−1 (We multiplied 3 and 2)={(11) of4}/2−1 (We added 6 and 5)={44}/2−1 (We wrote 11 of 4)= 22−1(We divided 44 and 2)=21(We subtracted 1 from 22)

Example 3:

What is the value of N = {(18 ÷ 3 × 2 + 1) of 8}/2 + 5?

Solution:

We will solve the terms inside the bracket first

{(18 ÷ 3 × 2 + 1) of 8}/2 + 5

= {6 × 2 + 1) of 8}/2 + 5 (We divided 18 and 3)

= {(12 + 1) of 8}/2 + 5 (We multiplied 6 and 2)

= {13 of 8}/ 2 + 5 (We added 12 and 1)

= (104)/2 + 5 (We wrote 13 of 8)

= 52 + 5 (We divide 104 and 2)

= 57 (We added 52 and 5)

Example4:What is the value of N= {(9 + 8÷ 2× 3) of8}/2 + 3 ?Solution:We will solve terms inside bracket first{(9 +8÷ 2 ×3) of8}/2 + 3={(9 + 4× 3) of8}/2 + 3(We divided 8 and 2 first)= {(9 + 12) of8}/2 + 3(We multiplied 3 and 4)= {(21) of8}/2 + 3(We added 9 and 12)= {168}/2 + 3(We wrote 21 of 8)= 84+ 3(We divided 168 and 2)= 87(We added 3 in 84 )">Example4:What is the value of N= {(9 + 8÷ 2× 3) of8}/2 + 3 ?Solution:We will solve terms inside bracket first{(9 +8÷ 2 ×3) of8}/2 + 3={(9 + 4× 3) of8}/2 + 3(We divided 8 and 2 first)= {(9 + 12) of8}/2 + 3(We multiplied 3 and 4)= {(21) of8}/2 + 3(We added 9 and 12)= {168}/2 + 3(We wrote 21 of 8)= 84+ 3(We divided 168 and 2)= 87(We added 3 in 84 )Example4:What is the value of N= {(9 + 8÷ 2× 3) of8}/2 + 3 ?Solution:We will solve terms inside bracket first{(9 +8÷ 2 ×3) of8}/2 + 3={(9 + 4× 3) of8}/2 + 3(We divided 8 and 2 first)= {(9 + 12) of8}/2 + 3(We multiplied 3 and 4)= {(21) of8}/2 + 3(We added 9 and 12)= {168}/2 + 3(We wrote 21 of 8)= 84+ 3(We divided 168 and 2)= 87(We added 3 in 84 )

__Fractions__

__The concept of fractions, though a simple one, can be often confused. Having not solved questions based on this simple concept, students often tend to confuse the problems. These questions can throw up the occasional challenge and it makes sense to practice these questions from this area.__

**Definition**: Technically, fraction is defined as part of the whole. The most common example of a fraction that comes to mind is half. When we say give me half of something, we are essentially demanding ½ part of it, in other words, ½ is the fractional representation for half.

Fractions are nothing else than the numerator divided by denominator, that is they occur in the form X/Y where X is the numerator and Y is the non-zero denominator.

**Remember:**

The numerator represents how many parts of that whole are being considered. To remember simply, numerator is the top number of the fraction that represents the numbers of parts that are to be chosen. The denominator represents the total number of parts created from the whole, in other words it is the bottom number representing the total number of parts created.

**Example of Fractions : **½, 2/3, 3/4, are the numbers which are in the form of x/y where y is non zero.

**Types of Fractions:**

**Proper Fraction: **When Numerator< Denominator, then the fraction is called as proper fraction**. **For example: 2/3, 4/5, 6/7 etc** **

**Improper fraction:** When Numerator >Denominator, then the fraction is called as improper fraction. For example**:** 5/3, 7/5, 19/7 etc

**Mixed fraction**: When a natural number combines with a fraction that is called a mixed fraction.

For Example: 2^{1}/_{2} ,3^{4}/_{5 }etc. In other words, the mixed fractions are improper fractions

**Tooltip: Properties of fractions**

** Property 1: **If we multiply the numerator and denominator by same quantity, the basic value of fraction will never change. For example

**:**4/5 x 5/5 = 20/25 = 4/5

** Property 2: **If there are two fractions a/b and c/d then a/b=c/d when ad=bc. For example 3/4 = 12/16 because 3 x 16 = 4 x 12

** Property 3: **A fraction with zero as the denominator is not defined.

** Property 4: **If the numerator of the fraction is zero, then the fraction equals to zero.

** Property 5: **If the numerator and denominator of the fraction are equal, then the fraction is equal to one.

**Basic Applications of Fractions****1. Fraction helps us determine the part of any number**

¾ part of 56 = ¾ x 56 = 42

4/5 part of 90 = 4/5 x 90 =72

**2. You can be asked to represent a number in the form of fraction.**

For example, you can be asked to represent 15 as a fraction of 450.

This can be done as follows:

15/450 = 1/30

We have solved the above example and it can be easily seen that 15 is our numerator and 450 is our denominator

**3. Always remember that the major quantity from which we have to extract something is the denominator.**

For example, when we say 4/5, we are essentially extracting four parts out of five.

**4. Extending the above concept, the quantity that is extracted is our numerator**

For example: 15/450 = 1/30

15 is the numerator because we have extracted 15 from 450 and the denominator is 450 because 15 is extracted from 450 so we can say that 1/30^{th} part of 450 is 15

** 5. ****Converting percentages into equivalent fractions makes our calculation easy**

For example:

If we remember 37.5% = 3/8 and one is asked to find 37.5% of 24 then we can write

37.5% of 24= 3/8 x 24 = 9** ****Remember few equivalent fractions for given percentages:**

50% = 1/2

25% = 1/4

12.5% = 1/8

6.25% = 1/16

33.33% = 1/3

66.66% = 2/3

75% = 3/4

20% = 1/5

40% = 2/5

60% = 3/5

80% = 4/5

16.66% = 1/6

83.33% = 5/6

14.28% = 1/7

28.57% = 2/7

42.85% = 3/7

57.14%= 4/7

71.42% = 5/7

85.71%=6/7

37.5% = 3/8

62.5% = 5/8

87.5% = 7/8

11.11% = 1/9

22.22% = 2/9

44.44% = 4/9

55.55% = 5/9

77.77% = 7/9

88.88% = 8/9

9.09% = 1/11

18.18% = 2/11

27.27% = 3/11

36.36% = 4/11

45.45% = 5/11

54.54% = 6/11

63.63% = 7/11

72.72% = 8/11

**DECIMALS**

Are all numbers integers? Well, the obvious answer to that question is a no. All numbers are not integers. Consider the case of 0.333333. What is this number? An integer? Well, it is a decimal. But what are decimals? Decimals are nothing else but the values lying between two integers on the number line.** Relating decimals to fraction: **When we solve a fraction of the form p/q where q is non-zero, it is not necessarily it would return an integral value. When we are left with a remainder, we ultimately convert it into decimal form. Some examples of Decimals are 4.5, 9.6, 6.78, and 99.98, these all are decimal numbers.

**Tooltip 1: Forming Decimals from Fractions**

How we can write 4/5 in decimal form?

Since the number 4 is smaller than the 5, so the decimal value will be less than one.

Multiple and divide both numbers by 10. We have:

40/ (10 x 5)

Which effectively is (dividing 40 by 5 first) 8/10

Thus, the final result is 0.8 So 4/5 = 0.8 and 0.8 is the decimal form of 4/5.

**Tooltip 2: Adding Decimals **

While adding decimals, you should always write the decimals in a vertical column with the decimal points aligned vertically.

Add all these 0.567 +78+8.9+5.06+56

= 78.000

+ 56.00

+ 5.06

+__0.567____ 139.627__

**Tooltip 3: Subtracting Decimal**

In addition we can write the numbers in any order. But while subtracting, we should preferably write the numbers in descending order and the vertical column with decimal points should be aligned to the same decimal points.

Let’s take an example: we have to subtract 0.567 from 5.06. If we write it as:

5.06__-0.567____ 4.507 __

This result is wrong because in case of subtraction we need equal digits in both the quantities, so these blank spaces are filled with 0

So this can be done like as

5.060__-0.567____ 4.493 __

This is the right approach for the question

**Tooltip 4: Multiplying Decimals**

As a first step, multiply the given integers in the normal form keeping the decimals aside. The number of decimal places in the product is then equal to the total of the decimal places in the two decimals. It is as simple as that.

Consider the following example:

5.060

x__0.567____2.869020__

First multiply 5060 with 567 and get the result as 2869020 and then move the decimal point to 6 places from the left i.e. between 2 and 8. Effectively, we move it to six decimals places, our sum total of the decimal places in the two numbers.

**Exponents**

Exponents are a method used to express products of the same number.

How would you write: 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2?

Hard to repeat the above number in calculations, isn’t it?

Well, exponents provide us the solution to the above problem. The above number simply becomes: 2^{9}

The expression A^{N} is called a power and it stands for A x A x A x A ……. x A , where there are N factors of A. A is called the base, and N is called the exponent. By definition, A^{0}= 1

In order to solve any question involving exponents you must remember the 5 basic rules of exponents. These 5 rules are universally applicable and help you solve any exponent problem with ease. These rules are:

*Also, keep in mind: a ^{-m} = 1/a^{m}*

Let’s use some of the rules above and try to solve some problems using the tips given above.

*Example 1: Which out of the following is equal to the expression 16*^{4}x 4^{3}?- 2
^{22} - 2
^{20} - 4
^{8} - 4
^{12} - 4
^{10}

*Solution:*

Let us reduce the expression to a common base. You should notice is that the number 16 can be expressed as 4^{2}. Thus, 16^{4} x 4^{3 } becomes ( 4^{2})^{4}

( 4^{2})^{4} is equal to 4^{2×4}, that is 4^{8}.

Thus, the original product becomes 4^{8} x 4^{3}= 4^{8+3} = 4^{11}= (2^{2})^{11} = 2^{22}

Thus, the correct answer is option A.*Example 2: If a not equal to zero, a(a ^{3})^{3}/a^{2} is equal to:*

- a
^{6} - a
^{7} - a
^{8} - a
^{9} - a
^{10}

*Solution:*

Let us first simplify the numerator.

a(a^{3})^{3}= a (a^{3×3}) =a x a^{9} = a^{10}

Now the equation becomes a^{10}/a^{2} = a^{8}

Thus, the correct answer is option C.

Divisibility Rules

A divisibility rule is a short cut for discovering whether a given number is divisible by a fixed divisor without performing the division, usually by examining its digits. Most of the divisibility rules are derived from a concept of remainders.

The two basic rules for finding out whether the number is divisible or not by any particular number:

1.If the divisor is prime number, then we can directly check for it.

2.If the divisor is a composite number then do factorization of divisor into factors, which are co-prime and check the divisibility for each prime factor individually

### Divisibility Rules: 2 to 7

### Divisibility Rules: 8 to 12

### Divisibility Rules: Special Numbers

Divisibility Rules explained with examples:

In this article, we use a few example to understand the concept of divisibility. Go through the following examples:

**Example 1: **

Check divisibility of 124 by 2, 3, 4 and 5.**Solution:**

124 is divisible by 2 since the last digit is 4.

124 is not divisible by 3 since the sum of the digits is 7 (1+2+4 = 7), and 7 is not divisible by 3.

124 is divisible by 4 since 24 is divisible by 4.

124 is not divisible by 5 since the last digit is 4 it is neither 0 nor 5.

124 is divisible by 2 and 4 not by 3 and 5.

**Example 2:**

Check divisibility of 225 by 2, 3, 4, 5, 6, 9 and 10.**Solution: **

225 is not divisible by 2 since the last digit is not 0, 2, 4, 6 or 8.

225 is divisible by 3 since the sum of the digits is 9 and 9 is divisible by 3.

225 is not divisible by 4 since 25 is not divisible by 4.

225 is divisible by 5 since the last digit is 5.

225 is not divisible by 6 since it is not divisible by 2.

225 is divisible by 9 since the sum of the digits is 9.

225 is not divisible by 10 since the last digit is not 0.

225 is divisible by 3, 5, 9.

**Example 3: **

Check divisibility of 400 by 2, 3, 4, 5, 6, 8, 9 and 10.**Solution:**

400 is divisible by 2 since the last digit is 0.

400 is not divisible by 3 since the sum of the digits is 4 and 4 is not divisible by 3.

400 is divisible by 4 since 00 is divisible by 4.

400 is divisible by 5 since the last digit is 0.

400 is not divisible by 6 since it is not divisible by 3.

400 is divisible by 8 since the last 3 digits are 400 and 400 is divisible by 8.

400 is not divisible by 9 since the sum of the digits is 4 and 4 is not divisible by 9.

400 is divisible by 10 since the last digit is 0.

400 is divisible by 2,4,5,8 and 10.

**Example 4:**

Check divisibility of 750 by 2, 3, 4, 5, 6, 8, 9 and 10.**Solution:**

750 is divisible by 2 since the last digit is 0.

750 is divisible by 3 since the sum of the digits is 12 and 1 is not divisible by 3.

750 is not divisible by 4 since 50 is not divisible by 4.

750 is divisible by 5 since the last digit is 0.

750 is divisible by 6 since it is divisible by 2 and 3.

750 is not divisible by 8 since the last 3 digits are 750 and 750 is not divisible by 8.

750 is not divisible by 9 since the sum of the digits is 12 and 12 is not divisible by 9.

750 is divisible by 10 since the last digit is 0.

750 is divisible by 2,3,5,6 and 10.

**Example 5: **

Check divisibility of 1200 by 2, 3, 4, 5, 6, 8, 9 and 10.**Solution:**

1200 is divisible by 2 since the last digit is 0.

1200 is divisible by 3 since the sum of the digits is 3.

1200 is divisible by 4 since 00 is divisible by 4.

1200 is divisible by 5 since the last digit is 0.

1200 is divisible by 6 since it is divisible by both 2 and 3.

1200 is divisible by 8 since the last 3 digits are 200.

1200 is not divisible by 9 since the sum of the digits is 3 and 3 is not divisible by 9.

1200 is divisible by 10 since the last digit is 0.

1200 is divisible by 2,3,4,5,6,8 and 10.

*Tricks for Divisibility** **Some basics tips and tricks that you can use for divisibility*

All whole numbers are divisible by 1.

A number is divisible by 2 if it’s even.

A non-zero number is divisible by 5 if it ends in 0 or 5.

In order to check the divisibility of a number by a composite number, divide the composite divisor into prime factors, which are co-prime and then check for its divisibility with each. For example, to check the divisibility of a number with 12, break down 12 into 3 and 4.

**Tricks for Divisibility****a ^{n} – b^{n} is always divisible by a-b**

**8**–

^{5}**5**is divisible by 8-5= 3

^{5}Remember it by:**a ^{3}**–

**b**is divisible by a-b

^{3}**a**–

^{2}**b**is also divisible by a-b

^{2}**a**

^{n}– b^{n}is divisible by a+b when n is even**7**is divisible by 7-5= 2

^{10}– 5^{10}Remember it by:

a

^{3}– b

^{3}is not divisible by a+b

a

^{2}– b

^{2}is also divisible by a+b

a

^{4}– b

^{4}is also divisible by a+b

**a ^{n} + b^{n }is divisible by a+b when n is odd**

**7**is divisible by 7+5= 12

^{11}+ 5^{11}Remember it by:

a

^{3}+ b

^{3}is divisible by a+b

a

^{2}+ b

^{2}is NOT divisible by a+b

a

^{4}+ b

^{4}is NOT divisible by a+b

**a ^{n} + b^{n}+c^{n}is divisible by a+b+c when n is odd.**

7

^{3}+ 5

^{3}+ 2

^{3}= 343+125+8=476 divisible by 7+5+2=14

*Questions you can solve by using above formula:***Example 1: 32 ^{23} + 17^{23}is definitely divisible by….**

a. 49

b. 15

c. 49 & 15

d. none of these.

ANSWER A

As

**a**

^{n}+ b^{n }is divisible by a + b when n is oddSo 32

^{23}+ 17

^{23}is divisible by 32 + 17 = 49.

**Example 2: 32 ^{23} – 17^{23} is definitely divisible by….**

a. 49

b. 15

c. 49 & 15

d. none of these.

ANSWER B

As

**a**

^{n}– b^{n}is always divisible by a-b.So 32

^{23}– 17

^{23}is divisible by 32 – 17 = 15.

**Example 3: 32 ^{232} – 17^{232} is definitely divisible by….**

a. 49

b. 15

c. 49 & 15

d. none of these.

ANSWER C

As **a ^{n} – b^{n} is always divisible by a-b** and

**a**

^{n}– b^{n}is divisible by a + b when n is evenSo 32

^{232}– 17

^{232}is divisible by both 32 – 17 = 15 and 32 + 17 = 49.

**Example 4: 3 ^{5} + 5^{5}+ 7^{5}is definitely divisible by….**

a. 8

b. 7

c. 15

d. all of these.

ANSWER C

As

**a**

^{n}+ b^{n }+ c^{n }is divisible by a + b + c when n is odd.So 3

^{5}+ 5

^{5}+ 7

^{5}is divisible by 3 + 5 + 7 = 15

**L.C.M. by Long Division Method**:

A least common multiple of two numbers is the smallest positive number that is a multiple of both.

Multiple of 3 — 3, 6, 9, 12, 15, 18,…………..

Multiple of 4 — 4, 8, 12, 16, 20, 24,………….

So the LCM of 3 and 4 is 12, which is the lowest common multiple of 3 and 4.

**An example of LCM**

The LCM of 10, 20,25 is 100. It means that 100 is the lowest common multiple of these three numbers, but there is a question in our mind that can the LCM be (-100)? Since (-100) is lower than 100 and divisible by each of 10, 20, 25, or can it be zero or what will be the LCM of (-10) and 20? Will it be (-20) or (-200)?

For all these questions, there is only one answer that the LCM is only defined for positive numbers and LCM is not defined for 0.

**PROCESS OF FINDING LCM**

- We will do prime factorization in first step of all the numbers.
- Then we calculate the number of times each prime occurs in prime factorization and write each number as power of primes.
- Then in last step we write all the primes involved and raise each of the primes to highest power present.

**Example based on above**

**Example 1:LCM of 10, 20, 25?****Step 1**: 10= 2 × 5

20 = 2 × 2 × 5

25= 5 × 5**Step 2****:** 10= 2^{1} × 5^{1}

20 = 2^{2} × 5^{1}

25= 5^{2}**Step 3****:**Primes involved are 2 and 5

Now we raise each of the primes to highest power present i.e.2^{2}^{ }× 5^{2} =100. So 100 is required LCM.

**Example 2: What is the LCM of 35, 45, 55?****Step-1: **

35 = 5 × 7

45 = 3 × 3 × 5

55 = 11 × 5**Step-2:**35 = 5

^{1}× 7

^{1}

45 = 3

^{2}× 5

^{1}

55 = 11

^{1}× 5

^{1}

**Step-3: Primes involved are 3, 5, 7 and 11**

Now we raise each of the primes to the highest power present i.e. 3

^{2}× 5

^{1}× 7

^{1}× 11

^{1}

LCM of the given numbers = 3465

**Another Method to calculate LCM**

Division method is another method to find the L.C.M. of numbers.

- First we will write the numbers in ascending order.
- Now we will divide the numbers with a common prime factor and continue the same till the time it is possible.
- We multiply all common prime factors and numbers obtained in last row to get the LCM.

Remember if a number is not divisible by this prime number, then write the number as it is

Let’s take an example to understand it:** ****Example 1: Find the L.C.M. of 24, 18 and 36?****Solution**:

Step 1: We arrange given numbers in ascending order i.e. we write as 18, 24 and 36.

Step 2: Here the common prime factors are 2 and 3. So we divide as shown in table below till the time it is possible.

Step 3: We multiply all common prime factors and numbers obtained in last row to get LCM= 2x2x3x3x2=72** ****Example 2: Find the L.C.M. of 60, 16 and 48?****Solution**:

Step 1: We arrange given numbers in ascending order i.e. we write as 16, 48 and 60.

Step 2: Here the common prime factors are 2 and 3. So we divide as shown in table below till the time it is possible.

Step 3: We multiply all common prime factors and numbers obtained in last row to get LCM= 2x2x3x4x5=240

**LCM of Fractions:**** **LCM of two or more fractions is given by as a fraction:

(LCM of numerators of all the fractions) / (HCF of denominator of all the fractions)

For example:

**LCM [ ½ , 5/12 ,7/6] = {LCM of (1,5,7)}/{HCF of (2,12,6)} = 32/2***Assignment:***1.**What is the LCM of 18 and 36?

A. 18

B. 36

C. 90

D. 72

**Answer:** Option **B****Explanation:**

18= 2x3x3=2^{1}x3^{2}

36=2x2x3x3=2^{2}x3^{2}

Primes involved are 2 and 3 and maximum power involved is 2 for both primes. Hence

2^{2 }x 3^{2} is required LCM.

**2.** What is the HCF of 18 and 36?

A.18

B.9

C.12

D.36

**Answer:** Option A**Explanation:**

18= 2x3x3=2^{1}x3^{2}

36=2x2x3x3=2^{2}x3^{2}

Primes involved are 2 and 3 and minimum power involved is 1 for both primes. Hence 2×3^{2}=18 is required HCF.

**3.** What is the LCM of 18, 180 and 216?

A. 540

B. 360

C. 720

D. 1080

**Answer:** Option D**Explanation**

18=2x3x3=2^{1}x3^{2}

180=2x2x3x3x5=2^{2}x3^{2}x5^{1}

72=2x2x2x3x3x3=2^{3}x3^{3}

Primes involved are 2, 3 and 5 and maximum powers involved are 3, 3 and 1 respectively. Hence 2^{3}x3^{3}x5^{1} = 1080 is required LCM

**4.** What is the HCF of 12,18 and 30?

A.10

B.5

C.6

D.8

**Answer:** Option C**Explanation: **

12=2x2x3x3=22×31

18=2x3x3=21×32

30 =2x3x5=21x31x51

Primes involved are 2, 3 and 5 minimum powers involved are 1 ,1 and 0 respectively. Hence

21x31x50=6 is required HCF.

5. What is the LCM of 13 and 11 ?

A.11

B.13

C.143

D.1

**Answer**: Option C**Explanation:**

Since both are prime numbers so LCM is nothing but their product = 13×11 = 143

**H.C.F. by Long Division Method**

To find the H.C.F. of the given number we will follow the following steps:

- We divide the bigger number by smaller one.
- Divide smaller number in step 1 with remainder obtained in step 1.
- Divide divisor of second step with remainder obtained in step 2.
- We will continue this process till we get remainder zero and divisor obtained in end is the required H.C.F.

Let’s take few examples for this:**Example 1**: Find the H.C.F. of 248 and 492?

To find the solution we will follow the following method i.e. we divide bigger number 492 by smaller one i.e. 248

So the divisor in the end was 4 so the H.C.F of the given numbers is 4**Example 2**: Find the H.C.F. of 420 and 396?

To find the solution we will follow the following method i.e. we divide bigger number 420 by smaller one i.e. 396

So the divisor in the end was 12 so the H.C.F of the given numbers is 12__Some points to remember__

– If H is the HCF of two numbers A and B, then H is also a factor of AX and BY, where X and Y are integers. In other words, H is also a factor of multiples of these numbers.

– If HCF (A,B) is H, then H is also the HCF of (A) and (A+B)

– If HCF (A,B) is H, then H is also the HCF of (A) and (A-B)

– If HCF (A,B) is H, then H is also the HCF of (A+B) and (A-B)

– If HCF=LCM for two numbers, then the numbers must be equal to each other.

– HCF of two or more fractions is given by:

(HCF of numerators of all the fractions) / (LCM of denominator of all the fractions)

**Shortcut for LCM **__Type 1:__**Remainder is same when divided by distinct divisors**

If the Number N leaves the same remainder R when divided by the numbers a, b, c, then the number N = (LCM of (a, b, c) p +R, where p is any whole number.

__Type 2:__**Difference between the divisors and remainders is same**

If the number N is divided by the numbers a, b, c and N gives three different remainders p, q, r respectively such that the difference between the divisors and remainders is same i.e.(a-p)=(b-q)=(c-r)=Z, then the number N must be in the form of {(LCM of a, b, c) p – Z} in each case, where p is any whole number.

__Type 3:__**Neither remainder is same nor is difference between the divisors and remainders same.**

When the number N is divided by the two numbers m and n and they leave remainders p, q respectively. In this case we identify the smallest number (say A) that satisfies given conditions (the smallest number which gives remainder p, q when divided by m, n and this number is identified by equating the number and using the fact that quotient is whole number). Once we have done that, the number must be in the form of {(LCM of a, b)p + A} (A is the smallest number)and p is any whole number.

**Let’s solve few examples based on above shortcuts:****Example 1: **Find the largest 4 digit number, which gives 4 as a remainder when divided by 17 and 19?** ****Solution:**

Let N be such number which satisfies given conditions.

Since N gives the same remainder 4 in each case. Hence N must be of the form

N = (LCM of 17, 19) p + 4where p is any whole number

Since LCM of 17 and 19 is 323 so the number must be in the form

N=323 p + 4

The largest value of “p” possible so that the N remains 4-digit number is 30.

Therefore, 323p = 323 X 30=9690.

Hence largest 4-digit number, which gives 4 as a remainder when divided by 17 and 19

is 9694.

**Example 2**: Find the largest 4-digit number, which gives the remainder 7 and 13 when divided by 11 and 17?**Solution: **

Let N be such number satisfying given conditions

Here remainders are not same but the difference between the remainders and the divisors is same i.e. 11-7 =17-13=4

Hence N must be of the form

N = (LCM of 11, 17) p – 4where p is any whole number

Since LCM of 11 and 17 is 187so the number must be in the form

N=187 p – 4.

The largest value of “p” possible so that the N remains 4-digit number is 53.

Therefore, 187p= 187 x 53=9911.

Hence the largest 4-digit number, which gives remainders 7 and 13 when divided by 11 and 17, is 9907.

**Example 3**: Find the largest 3-digit number, which gives the remainder 1 and 2 when divided by 3 and 5?**Solution: **

Let N be such number satisfying given conditions

i.e. N = 3a + 1 where a is any whole number

&N = 5b + 2 where b is any whole number.

Where a & b are whole numbers.

Equating

3a + 1=5b + 2

b= (3a-1)/5

Since b is whole number. So, 5 should divide numerator implies a=2.

Thus smallest number satisfying both given conditions is 7.

So the number will be in the form of (LCM of 3,5)k + 7.

Now we have to find the largest 3-digit number N in the form of 15k+7

The largest value of “k” possible so that the N remains 3-digit number is 53.

Hence the largest 3-digit number, which gives the remainder 1 and 2 when divided by 3 and 5 is 15 x 66 + 7 = 997

**Shortcuts for HCF:****There are two types of question usually asked in the examination based on H.C.F.**

**TYPE – 1**

When various number a, b, c is divided by number N and gives the same remainder r in each case. Then H.C.F. of a, b, c is given by the H.C.F. of (a – b) or (b – a) and (b–c) or (c – b) and (c –a) or (a –c). Let’s understand with help of example

**Example 1**: Find the largest possible value of N if all three numbers 59, 77,104 when divided by N leaves same remainder?**Solution:** As per given

59 = N x a + r………………………..(1)

77 = N x b + r………………………..(2)

104 =N x c + r………………………..(3)

Subtract equation (1) from (2)

77 – 59 =18 = N x (b -a)……………………..(4)

Subtract equation (2) from (3)

104 – 77=27 = N x (c – b)…………………….(5)

Subtract equation (1) from (3)

104 -59 =45 = N x (c – a)……………………..(6)

From this we can conclude that the N has to be the factor of 18, 27,45

Now we are asked for the largest number so we will take the H.C.F. of 18, 27, 45 and which is 9 .So 9 is the largest number which divides 59, 77, 104 and gives same remainder in each case.

**TYPE – 2**

When the number is divided by various numbers a, b, c and gives different remainders p, q, r.

**Example 2: **Find the largest number N, which divides, 162,292,325 and gives remainders 1, 5 and 3 respectively?**Solution: **As per given information

162 = N x a + 1………………………..(1)

292 = N x b + 5………………………..(2)

325 = N x c + 3………………………..(3)

Since the three numbers are 162, 292 and 325 and the remainders are 1, 5, 3 so we can say that the number is totally divisible with 161, 287, 322. So finding largest number, which divides 162,292,325, is equivalent to finding H.C.F of 161, 287 and 322.Since HCF (161, 287, 322)=7 so the 7 is the number which when divided by the 162,292,325 gives the remainder 1,5 and 3 respectively.

**Some points to remember **

1. LCM x HCF of two numbers is equal to the product of numbers.

2. If HCF (H) of two numbers is given, then the numbers can be assume as HCF x A and HCF x B (A and B are integers)

3. HCF of a given set of numbers is always a factor of LCM

**Basics of Factors:**

What are factors? What does the term mean? If we go by the Latin translation of the word, it means ‘who/which acts’. Well, in our mathematical sense, factors are the ones that act for sure, but in this case they act on numbers.

** Mathematically speaking:** If any integer, say P, is divisible by another integer say Q an exact number of times then P is said to be a multiple of Q and Q is the factor of P.

**Number of Factors of a given number:**

Number of factors can be expressed by following steps:

- First write down the number in prime factorisation form i.e. a
^{p}b^{q}c^{r }(where a,b,c, are prime numbers and the p,q,r are natural numbers as their respective powers) - Number of factors can be expressed as (p+1)(q+1)(r+1).
- Here factors include 1 and the number itself.

**The points above explained through a example:**

Let us take an example of a number N = 38491200=2^{6} 3^{7} 5^{2}

Now observe some facts about the number of factors (we will solve the problem step by step):**Step 1:** Prime factorisation, so N=38491200=2^{6} 3^{7} 5^{2} 11

Power of 2 as 2^{0 }, 2^{1} ,2^{2 } ,2^{3},2^{4},2^{5},2^{6}( 6+1=7)ways ,

Power of 3 as 3^{0 }, 3^{1} ,3^{2 },3^{3},3^{4,},3^{5},3^{6},3^{7}( 7+1=8)

Power of 5 as 5^{0 }, 5^{1} ,5^{2 }( 2+1=3)ways

Power of 11 as 11^{0 }, 11^{1} ^{ }( 1+1=2)ways**Step 2:**Hence, the number of factors is given by (6+1)(7+1)(2+1)(1+1)=7x8x3x2=336

**Example 1**:Find the **number of factors** of 2^{4}3^{1}5^{2}7^{2}.**Solution:** As we can see the above number has 2,3,5,7 which all are prime numbers and they have 4,1,2,2 as their powers so the number of factors of the given number are (4+1)(1+1)(2+1)(2+1)= 90** ****Example 2**:Find the **number of factors** of 1440 ?**Solution:** We first factorize 1440.

1440 = 2^{5}3^{2}5^{1}

2,3,5 are prime numbers and they have 5,2,1 as their powers so the number of factors of the given number are (5+1)(2+1)(1+1)= 36

**Various Types of Factors**** Even factors**:

Even factors are the factors of number, which are divisible by 2.

**Example 1**: **Find the number of even factors of 58800?****Solution**: We first factorize 58800.

58800 = 2^{4 }3^{1}5^{2}7^{2}

In this case we have to find number of even factors, an even factor is divisible by 2 or we can say smallest power of 2 should be 1 not 0.

Hence a factor must have

2^{(1 or 2 or 3 or 4) } — 4 factors

3^{(0 or 1 ) } — 1+1=2 factors

5^{(0 or 1 or 2)} — 1+2=3 factors

7^{(0 or 1 or 2) }— 1+2=3 factors

Hence total number of even factors = (4)(2)(3)(3) = 72**Hence, number of even factors of a number **

**N=(2)^p*(a)^q*(b)^r*(c)^s are p(q + 1)(r + 1)(s + 1)**

__Odd factors__**: **

Odd factors are those factors, which are not divisible by 2.

**Example 2**: **Find the number of odd factors of 58800.****Solution: **We first factorize 58800.

58800 = 2^{4 }3^{1}5^{2}7^{2}

Total number of factors of 58800 is 5x2x3x3=90 and in previous example we calculated total number of even factors is 72.

Number of odd factors = Total number of factors – Number of even factors

=90- 72 = 18

** Alternate way**: Since odd factors should have power of 2 as 0. Hence odd factor must have

2

^{(0 ) }———- 1 factor

3^{(0 or 1 ) } — 1+1=2 factors

5^{(0 or 1 or 2)} — 1+2=3 factors

7^{(0 or 1 or 2) }— 1+2=3 factors

Total number of odd factors = (1 )(2)(3)(3) = 18

Hence, number of odd factors of a number

N=2^(p)*a^(q)*b^(r)*c^(s)are p(q + 1)(r + 1)(s + 1)

__Perfect square factors:__

If a number is perfect square then its prime factors must have even powers.

**Example 3: Find the number of factors of 58800 which are perfect square?****Solution:**

We know that for a number to be a perfect square, its factor must have the even number of powers.

We first factorize 58800.

58800 = 2^{4 }3^{1}5^{2}7^{2}

Hence perfect square factors must have

2^{(0 or 2 or 4)}—– 3 factors

3^{( 0 ) } —– 1 factor

5^{(0 or 2)} ——- 2 factors

7^{(0 or 2) }— 2 factors

Hence, the number of factors which are perfect square are 3 x 1 x 2 x 2=12

**Remember ****-If number of factors is odd then the number is a perfect square and vice versa is also true i.e. if a number is a perfect square then number of factors is odd.****This is because if number is a perfect square then p, q, and r are even and hence ****(p + 1) (q + 1) and (r+ 1) are odd and so product of these numbers is also an odd number. ****-If number of factors is even then number is not a perfect square.**

__Perfect cube factors:__

If a number is a perfect cube, then the power of the prime factors should be divisible by 3.**Example 1:Find the number of factors of2 ^{9}3^{6}5^{5}11^{8} that are perfect cube?**

**Solution:**If a number is a perfect cube, then the power of the prime factors should be divisible by 3. Hence perfect cube factors must have

2

^{(0 or 3 or 6or 9)}—– 4 factors

3

^{(0 or 3 or 6)}—– 3 factors

5

^{(0 or 3)}——- 2 factors

11

^{(0 or 3 or 6 )}— 3 factors

Hence, the total number of factors which are perfect cube 4x3x2x3=72

__Perfect square and perfect cube__If a number is both perfect square and perfect cube then the powers of prime factors must be divisible by 6.

**Example 2**:

**How many factors of 2**

^{9}3^{6}5^{5}11^{8}are both perfect square and perfect cube?**Solution:**If a number is both perfect square and perfect cube then the powers of prime factors must be divisible by 6.Hence both perfect square and perfect cube must have

2

^{(0 or 6)}—– 2 factors

3

^{(0 or 6)}—– 2 factors

5

^{(0)}——- 1 factor

11

^{(0 or 6)}— 2 factors

Hence total number of such factors are 2x2x1x2=8

**Example 3: How many factors of2**

^{9}3^{6}5^{5}11^{8}are either perfect squares or perfect cubes but not both?**Solution:**

Let A denotes set of numbers, which are perfect squares.

If a number is a perfect square, then the power of the prime factors should be divisible by 2. Hence perfect square factors must have

2

^{(0 or 2 or 4 or 6 or 8)}—– 5 factors

3

^{(0 or 2 or 4 or 6)}—– 4 factors

5

^{(0 or 2or 4 )}——- 3 factors

11

^{(0 or 2or 4 or6 or 8 )}— 5 factors

Hence, the total number of factors which are perfect square i.e. n(A)=5x4x3x5=300

Let B denotes set of numbers, which are perfect cubes

If a number is a perfect cube, then the power of the prime factors should be divisible by 3. Hence perfect cube factors must have

2

^{(0 or 3 or 6or 9)}—– 4 factors

3

^{(0 or 3 or 6)}—– 3 factors

5

^{(0 or 3)}——- 2 factors

11

^{(0 or 3 or 6 )}— 3 factors

Hence, the total number of factors which are perfect cube i.e. n(B)=4x3x2x3=72

If a number is both perfect square and perfect cube then the powers of prime factors must be divisible by 6.Hence both perfect square and perfect cube must have

2

^{(0 or 6)}—– 2 factors

3

^{(0 or 6)}—– 2 factors

5

^{(0)}——- 1 factor

11

^{(0 or 6)}— 2 factors

Hence total number of such factors are i.e.n(A∩B)=2x2x1x2=8

We are asked to calculate which are either perfect square or perfect cubes i.e.

n(A U B )= n(A) + n(B) – n(A∩B)

=300+72 – 8

=364

Hence required number of factors is 364

**Product of Two factors ****To find the number as the products of two factors, use the following steps :****Step1**: Write Prime factorisation of given number i.e. convert the number in the form a^{p} b^{q} c^{r}

where a ,b,c are prime numbers and the p,q,r are natural numbers as their respective powers.**Step 2**:Find Number of factors which can be expressed as( p+1)(q+1)(r+1).**Step 3**: Number of ways to express the number as a product of two numbers is exactly half its number of factors i.e.½ *(p+1)(q+1)(r+1).**Let’s have an example on this** :**Example 1**:** In how many ways can you express 54 as a product of two of its factors? ****Solution**: We will do the above problem step by step:

Step 1: Prime factorization of 54 i.e. we write 54 = 2^{1}3^{3}

Step 2: Number of factors of 54 will be (1+1)(3+1) = 2 x 4= 8

Step 3: Hence number of ways to express 54 as a product of two numbers is exactly half its number of factors i.e. ½ *8 = 4 ways.

In fact we can list these 4 ways as well

Factors of 54 are 1,2,3,6,9, 18,27,54.

Now it is very simple to find the factors from 1 to 9 but it is difficult to find the ones that are greater than 10. So number of ways to express 54 as a product of two of its factors is

1 x 54 = 54

2 x 27 = 54

3 x 18 = 54

6 x 9 = 54**Example 2**: **In how many ways you can express 120 as a product of two of its factors? ****Solution**:

Step 1: Prime factorization of 120 i.e. we write 120 = 2^{3}3^{1}5^{1}

Step 2: Number of factors of 120 will be (3+1)(1+1)(1+1) = 4 x 2 x2 = 16

Step 3: Hence number of ways to express 120 as a product of two numbers is exactly half its number of factors i.e. ½ *16 = 8

In fact we can list these 8 ways as well

Factors of 120 are 1,2,3,4,5,6,8,10,12,15,18,24,30,40,60,120.

So number of ways to express 120 as a product of two of its factors is

1 x 120 = 120

2 x 60 = 120

3 x 40 = 120

4 x 30 = 120

5 x 24 = 120

6 x 20 = 120

8 x 15 = 120

10 x 12=120

__Product of Factors____Perfect square as a product of two factors__

In case of perfect square number we have odd number of factors i.e. the number of factors are odd hence in that case required number of ways in which we can write perfect square number as a product of its two factors are (n – 1)/2 if we do not include the square root of the number and

(n + 1 )/2 if we include the square root of the number. So number of ways to express a perfect square as product of two different factors (that means excluding its square root) is

½ {(p + 1)(q + I)(r + I) … – 1)}. And if we include the square root then required numbs 1/2 {(p+1)(q+1)(r+1) … +1}**Let’s take one example to understand this. ****Example 2**: **In how many ways you can express 36 as the product of two of its factors?****Solution :****Step 1**: Prime factorization of 36 i.e. we write 36 = 2^{2} x 3^{2}**Step 2**: Number of factors of 36 will be (2+1)(2+1)=9 (i.e. factors are 1,2,3,4,6,9,12,18,36)**Step 3**: Since we are asked total number of ways hence we include square root of 36 i.e. 6 as well. Thus number of ways you can express 36 as the product of two of its factors is (9+1)/2=5__ ____Product of two co-prime numbers __

To express the number as a product of co-prime factors we will use the following steps:**Step 1:**Write Prime factorisation of given number i.e. convert the number in the form where p_{1} ,p_{2},p_{3}…..p_{n }are prime numbers and a,b,c….. are natural numbers as their respective powers.**Step 2:** In the above step we have n prime numbers then the number of ways to express the number as the product of two co prime numbers =2^{n-1}

Because two primes are always co-prime and after we pick 1 prime the other prime can be picked in 2^{n-1}ways.Hence number of ways in which we can write given number as a product of two co prime factors =2^{n-1}**Example 1: In how many ways you can write 315 as product of two of its co-prime factors. Solution: **

Step 1: Prime factorization of 315 i.e. we write 315 = 3

^{2}x 5

^{1}x 7

^{1}

Step2: In above step 3 prime numbers are used. Hence number of ways are 2

^{3-1 }= 4.Infact we can mention these cases as well 9×35, 5 x 63, 7 x 45, 15 x 21.

__Remember:__The number of numbers, which are less than N=**p**

^{a}q^{b}r^{c}(where p , q, r are prime numbers and the a,b,c are natural numbers as their respective powers)and are co-prime to N is given by N{(1-**1/p)(1-1/q**

**)(1-1/r**

**)}.**

__Product of all the factors__**To find product of all the factors we follow the following steps:**

Step1: Prime factorisation N= a

^{p}b

^{q}c

^{r}

Step2: Number of factors (say X)

Step3: Product of all the factors is given by = N

^{X/2}

**Example 2: Find the product of all the factors of 120 ?**

**Solution:**

**Step 1:**Prime factorisation : = 2

^{3}X5

^{1}X3

^{1}

**Step2:**Number of factors are ( 3+1)(1+1)(1+1)= 16

**Step 3:**Product of all the factors =( 2

^{3}X5

^{1}X3

^{1})

^{16/2}=(120)

^{8}

*Assignment:*

__Questions:__**1**: In how many ways can you express 216 as a product of two of its factors?

a) 4

b) 6

c) 8

d) 9

Answer: c

Solution:

Step 1: 216 = 2

^{3}3

^{3}

Step 2: Number of factors are (3+1)(3+1) = 4 x 4= 16

Step 3: So number of ways ½ (4)(4) = 8

**2**: Find the sum of factors of 216.

a. 950

b) 850

c) 600

d) 1000

Answer: c

Solution:

Step 1: 216 = 2^{3}3^{3}

Step 2: Sum of factors = (2^{0}+2^{1}+2^{2}+2^{3})(3^{0}+3^{1}+3^{2}+3^{3})= 15 x 40 = 600

**3**: Find the product of all the factors of 400

a. 80^{5}

b) 80^{4}

c) 80^{7}

d) 80^{6}

Answer: a

Solution:

Step 1: Prime factorisation of 400 = 2^{4}X5^{2}

Step2: Number of factors (4+1)(1+1)= 10

Step 3: Product of all the factors =( 2^{4}X5^{1})^{10/2} =(80)^{5}

**4**:**In how many ways you can write 200 as product of two of its co-prime factors.**

a) 1

b) 2

c) 4

d) 8

Answer: b

Step1: 200 = 2

^{3}5

^{2}

Hence number of ways are 2

^{2-1 }= 2.Infact we can mention these cases as well 8 x 25,

1 x 100.

**5**: In how many ways you can write 10890 as product of two of its co-prime factors.

a) 7

b) 3

c) 31

d) 15**Answer: d Solution: **We will do the solution with step by step

Step1: first prime factorization i.e.2

^{1}x 3

^{2}x 5

^{1}x 11

^{2}

Step2: 3 prime number are used in this so the number of ways are 2

^{4}-1=15

**Basic Concept of Cyclicity**

The concept of cyclicity is used to identify the last digit of the number.Let’s take an example to understand this:

**Example 1: Find the unit digit of 3 ^{54}.**

**Solution:**Now it’s a very big term and not easy to calculate but we canfind the last digit by using the concept of cyclicity. We observe powers of 3

3

^{1}= 3

3

^{2}= 9

3

^{3}= 27

3

^{4}= 81

3

^{5 }= 243

3

^{6}= 729

So now pay attention to the last digits we can observe that the last digit repeats itself after a cycle of 4 and the cycle is 3, 9, 7, 1 this repetition of numbers after a particular stage is called the cyclicity of numbers. Therefore when we need to find the unit digit of any number like 3^{n }we just need to find the number on which the cycle halts. So we divide power n by 4 to check remainder

- If remainder is 1 then the unit digit will be 3
- If remainder is 2 then the unit digit will be 9
- If remainder is 3 then the unit digit will be 7
- If remainder is 0 then the unit digit will be 1

We divided the power by 4 because cycle repeats itself after 4 values .Now the main question was that how much is the last digit of 3^{54 }and we know the cycle repeat itself after 4 so we will divide the 54 with 4,so on dividing 54 by 4 the remainder becomes 2 .Now as we discussed above if the remainder is 2 the last digit would be 9.Hence unit digit of 3^{54 }is 9.

**Example 2:What will be the unit digit of 347 ^{45}**

**Solution:**

Lets observe unit digit of 347 x 347 = 9.

The main purpose of the above is that the unit digit of any multiplication depends upon the unit digit of numbers, whatever is the number big or small the unit digit always depends upon the multiplication of the last digit.

So the last digit of 347

^{45}can be found by calculating last digit of 7

^{45}

We observe unit digit while calculating powers of 7

7

^{1}= 7

7

^{2}= 49

7

^{3}= 343

7

^{4}= 2401

7

^{5}= 16807

So on dividing 45 with 4, 1 will be the remainder and the last digit would be 7.

**Application of Concept of Cyclicity**

__How to calculate unit digit if a number contains power of a power__

**Example 1:What will be the last digit of 12^{(23)^(45)}**

**Solution:**

To find the last digit of this type of number we will start the question from the base the base is given to be 12. It means we will see the cyclicity of 2 because the last digit depends upon the unit digit of 12 i.e. 2. We observe unit digit while calculating powers of 2

2^{1} = 2

2^{2} = 4

2^{3} = 8

2^{4} = 6

2^{5} = 2

**Step 1**: Now we know that cyclicity of last digit of 12 i.e. 2 is of 4, hence we divide the power of 12 i.e. 23^{45} with 4.

**Step 2**: Now lets calculate the remainder of 23^{45 }when divided by 4 and then we will determine the last digit.

**Step 3**: The remainder will be 3 because we can write remainder of 23 by 4 as 3 or -1.Hence we can write 23^{45} as (-1) ^{45} but odd power of -1 will be again -1 and thus 23^{45 }when divided by 4 will give us remainder as -1 or 3.

Hence unit digit of **12^{(23)^(45)}** will be same as unit digit of 2^{3}i.e. 8

**Assignment:**__Questions:__**1:** Find the unit digit of 4^{686}

a) 4

b) 6

c) 2

d) 3

Answer: b

Solution: We know

4^{1} = 4

4^{2} = 6

4^{3} = 4

4^{4} = 4

Here, we see that powers of four repeat after a cycle of 2 i.e.

Odd power of 4 gives unit digit as 4 and even power of 4 gives unit digit as 6

So unit digit of 4^{686 }is 6.

**2 :** What will be the unit digit of 65^{1234}

a) 5

b) 0

b) 2

d) 3

Answer: a

Solution We know

5^{1} = 5

5^{2} = 25

5^{3} = 125

5^{4} = 625

So, we observe that unit digit of any power of 5 is 5.Hence unit digit of 65^{1234} is 5.

**3:** Find the unit digit of 36^{512}

a) 4

b) 6

c) 2

d) 3

Answer: b

Solution We know

6^{1} = 6

6^{2} = 36

6^{3} = 216

So, we observe that unit digit of any power of 6 is 6.Hence unit digit of 36^{512} is 6.

**4:** Find the unit digit of 18^{5693}

a) 4

b) 6

c) 2

d) 8

Answer: d

Solution: We observe unit digit while calculating powers of 8

Unit digit in 8^{1} = 8

Unit digit in 8^{2} = 4

Unit digit in 8^{3} = 2

Unit digit in 8^{4} = 6

Unit digit in 8^{5} = 8

So on dividing 5693 with 4, 1 will be the remainder and hence the last digit would be 8.

### How to find the Unit Digit of a number

For the concept of identifying the unit digit, we have to first familiarize with the concept of cyclicity. Cyclicity of any number is about the last digit and how they appear in a certain defined manner. Let’s take an example to clear this thing:

The cyclicity chart of 2 is:

2^{1 }=2

2^{2} =4

2^{3 }=8

2^{4}=16

2^{5}=32

Have a close look at the above. You would see that as 2 is multiplied every-time with its own self, the last digit changes. On the 4^{th} multiplication, 2^{5} has the same unit digit as 2^{1}. This shows us the cyclicity of 2 is 4, that is after every fourth multiplication, the unit digit will be two.

Cyclicity table:

The cyclicity table for numbers is given as below:

How did we figure out the above? Multiply and see for yourself. It’s good practice.

Now let us use the concept of cyclicity to calculate the Unit digit of a number.

**What is the unit digit of the expression 4 ^{993}?**Now we have two methods to solve this but we choose the best way to solve it i.e. through cyclicity

We know the cyclicity of 4 is 2

Have a look:

4

^{1}=4

4

^{2 }=16

4

^{3 }=64

4

^{4 }=256

From above it is clear that the cyclicity of 4 is 2. Now with the cyclicity number i.e. with 2 divide the given power i.e. 993 by 2 what will be the remainder the remainder will be 1 so the answer when 4 raised to the power one is 4.So the unit digit in this case is 4.

For checking whether you have learned the topic, think of any number like this, calculate its unit digit and then check it with the help of a calculator.

* Note* :

*If the remainder becomes zero in any case then the unit digit will be the last digit of*

**a**^{cyclicity number }*where a is the given number and cyclicity number is shown in above figure.*

Lets solve another example:**The digit in the unit place of the number 7 ^{295} X 3^{158} is**

A. 7

B. 2

C. 6

D. 4

**Solution**

The Cyclicity table for 7 is as follows:

7^{1 }=7

7^{2 }=49

7^{3} = 343

7^{4} = 2401

Let’s divide 295 by 4 and the remainder is 3.

Thus, the last digit of 7^{295} is equal to the last digit of 7^{3} i.e. 3.

The Cyclicity table for 3 is as follows:

3^{1} =3

3^{2} =9

3^{3 }= 27

3^{4 }= 81

3^{5 }= 243

Let’s divide 158 by 4, the remainder is 2. Hence the last digit will be 9.

Therefore, unit’s digit of (7^{925} X 3^{158}) is unit’s digit of product of digit at unit’s place of 7^{925} and 3^{158} = 3 * 9 = 27. Hence option 1 is the answer.

### Factorials

A factorial is a non-negative number which is equal to the multiplication of numbers that are less than that number and the number itself. It is denoted by (!)**Let’s take an example to understand this **

What will be the value of 5!

So in the above definition we discussed that the multiplication of the numbers which all are less than that number and the number itself. Hence number less than 5 are 1,2,3,4 and 5 is number itself so

5! = 5 x 4 x 3 x 2 x 1 = 120

Always remember we define the value of 0! =1

**Lets take some more examples for this **

Value of 6! = 6 x 5 x 4 x 3 x 2 x 1 = 720

Value of 4! = 4 x 3 x 2 x 1 = 24

Value of 3! = 3 x 2x 1= 6

We can also write n! = n x (n – 1)!

Concept of factorial is very important, there are few basic types in which this concept is used. We will discuss each type with the help of examples.** ****Type 1: Highest power of p (p is a prime number) which divides the q! **

Let’s take an example to understand this type** ****What is the highest power of 3 that divides 13! ?****Solution**:

13! = 13 x 12 x 11 x 10 x 9 x 8 x7 x 6 x 5 x 4 x 3 x 2 x 1

And we have to find highest power of 3 that can divide the above term

So first we need to know how many times 3 is multiplied in 13! .

So to find the number of 3’s we will divide 13 with 3.

13 divided by 3 gives 4 as quotient and 1 remainder. We will keep remainder aside and move further till the quotient cannot be divided further

Again we divide 4 by 3 we get quotient as 1 and again remainder is 1 .Now we stop at this stage because quotient 1 cannot be divided by 4.

Now add all the quotients

4+1 =5.

So the maximum power of 3 is 5, which can divide the 13! .

**We can also generalize the formula for calculating **

Maximum power of any prime p in any n! is calculated as

[np]+[np2]+[np3]+[np4]+………………">[np]+[np2]+[np3]+[np4]+………………[np]+[np2]+[np3]+[np4]+………………

Where [ ] represents integral part

Maximum power of 3 in 13!

[133]+[1332]+………..">[133]+[1332]+………..[133]+[1332]+………..

= 4+ 1 =5

**Other method**

Since the number 13! is not very big number in-fact we can write and check maximum power of 3

13 x 12 x 11 x 10 x 9 x 8 x7 x 6 x 5 x 4 x 3 x 2 x 1

13 x **3 x4** x 11 x 10 x **3 x 3** x 8 x 7 x **3** x 2 x **3** x 2 x 1

Here the number of 3’s is 5 so we can cancel it with 3^{5}. So maximum power of 3 is 5, which can divide 13!

**Let’s take one more example like this ****What is the highest power of 7 that exactly divides 49! **

To find the highest power of 7 that exactly divides 49! . We need to know the number of 7’s in the 49!

So when we divide 49 with 7, the quotient will be 7 and there is no remainder, since the quotient can further divided by 7 we will divide 7 with 7 with quotient 1 and remainder 0 .

Now add all quotients and get the answer as 8

So the highest power which will divide the 49! will be 8.

## Highest Power of a prime in a factorial

**Type : Highest power of p which divides the q! ,where p is not a prime number**

The approach for this type is same as that for calculating maximum power of prime in any factorial buthere first we will break p into product of primes.

Lets take an example to understand this

__Example 1__**What will be the maximum power of 6 that divides the 9!**

In order to find maximum power of 6 we will first write as product of 2 and 3.

Maximum power of 2 in 9!

[92]+[922]+[923]+…..= 4 + 2 + 1 + 0=7Maximum power of 3 in 9![93]+[932]+…..= 3+ 1+0=4">[92]+[922]+[923]+…..= 4 + 2 + 1 + 0=7Maximum power of 3 in 9![93]+[932]+…..= 3+ 1+0=4[92]+[922]+[923]+…..= 4 + 2 + 1 + 0=7Maximum power of 3 in 9![93]+[932]+…..= 3+ 1+0=4

And there are seven 2’s and four 3’s are there, so we need to make the pairs of 2 and 3.

Now we know that 6 = 2 x3. So we need equal number of 2 and equal number of 3 in 9!

Hence there are maximum four pairs that can make 6.So the maximum power of 6 that

can divide 9! is 4.

**Other method**

Since the number 9! is not very big number in-fact we can write and check maximum power of 3

9! = 9x8x7x6x5x4x3x2x1 =**3×3**x**2x2x2**x7x**2×3**x5x**2×2**x**3**x**2**x1

So there are four pairs of 2 x 3, which can be formed

So the maximum power of 6 that can divide the 9! is 4 .

__Example 2__**What will be the highest power of 12 that can exactly divide 32!**

We can write 12 = 2x2x3 i.e. we need pair of 2^{2} x 3

Maximum power of 2 in 32!

[322]+[3222]+[3223]+[3224]+[3225]+…..= 16 + 8 + 4 + 2 +1=31Maximum power of 3 in 32![323]+[3232]+[3233]+…..= 10+ 3+ 1= 14">[322]+[3222]+[3223]+[3224]+[3225]+…..= 16 + 8 + 4 + 2 +1=31Maximum power of 3 in 32![323]+[3232]+[3233]+…..= 10+ 3+ 1= 14[322]+[3222]+[3223]+[3224]+[3225]+…..= 16 + 8 + 4 + 2 +1=31Maximum power of 3 in 32![323]+[3232]+[3233]+…..= 10+ 3+ 1= 14

Now in 32! Number of 2 are 31

And the number of three’s = 14

So therefore if we take 14 three’s then we need 28 two’s because for each three we need two 2’s so therefore we need 28 two’s from 31 two’s to make the equal pairs of 2^{2} x3

So the maximum power of 12 that can divide the 32! is 14

### Number of zeros at the end of a factorial

**Number of zeros at the end of a ****Problem Type: Number of zeros at the end of p!**

To solve such type of problems we see only pair of 2 x5 in p! .

Because if a number is divisible by 10 then it will have 0 in the end and 10= 2×5 so finding number of zeros is equivalent to finding maximum power of 10 in p! which is same as counting number of pairs of 2 and 5.

Lets take an example for this

__Example__**Find the number of zeros at the end of 45! ****Solution:**

Zero mainly comes from the combination of (5x 2) or by the presence of 10, and the number of zeros depends upon the number of times 10 is involved.So we check power of 2 and 5 first

Maximum power of 2 in 45!

[452]+[4522]+[4523]+[4524]+[4525]+…..= 22 + 11 + 5 + 2 +1= 41Maximum power of 5 in 45![455]+[4552]+[4553]+…..= 9+ 1= 10">[452]+[4522]+[4523]+[4524]+[4525]+…..= 22 + 11 + 5 + 2 +1= 41Maximum power of 5 in 45![455]+[4552]+[4553]+…..= 9+ 1= 10[452]+[4522]+[4523]+[4524]+[4525]+…..= 22 + 11 + 5 + 2 +1= 41Maximum power of 5 in 45![455]+[4552]+[4553]+…..= 9+ 1= 10

So in 45! Number of 2’s are 41

And the number of 5’s is 10

So maximum pair of 2 and 5 that can be made are 10 so the number of zeros at the end of the 45! is 10.

__Example __**Find the number of zeros in 500!****Solution: **Zero mainly comes from the combination of (5x 2) or by the presence of 10, and the number of zeros depends upon the number of times 10 is involved.So we check power of 2 and 5 first

Maximum power of 2 in 500!

[5002]+[50022]+[50023]+[50024]+[50025]+[50026]+[50027]+[50028]…= 250 + 125 + 62 + 31 +15 +7 + 3 + 1= 494Maximum power of 5 in 500![5005]+[50052]+[50053]+[50054]+…..= 100 + 20 + 4= 124">[5002]+[50022]+[50023]+[50024]+[50025]+[50026]+[50027]+[50028]…= 250 + 125 + 62 + 31 +15 +7 + 3 + 1= 494Maximum power of 5 in 500![5005]+[50052]+[50053]+[50054]+…..= 100 + 20 + 4= 124[5002]+[50022]+[50023]+[50024]+[50025]+[50026]+[50027]+[50028]…= 250 + 125 + 62 + 31 +15 +7 + 3 + 1= 494Maximum power of 5 in 500![5005]+[50052]+[50053]+[50054]+…..= 100 + 20 + 4= 124

Number of 5’s in 500! are 124

And the numbers of 2’s is 494

So the maximum possible pairs of 2 and 5 that can be made are 4 so the number of zeros in 500! are 124 .So the number of zeros in the end of the 500! are 124.

__Example__**What will be the number of zeros in (24!) ^{3! }?**

Now first calculate the number of zeros in 24!

Maximum power of 2 in 24!

[242]+[2422]+[2423]+[2424]+…..= 12 + 6 + 3 + 1= 22Maximum power of 5 in 24![245]+[2452]+…..= 4 + 0= 4">[242]+[2422]+[2423]+[2424]+…..= 12 + 6 + 3 + 1= 22Maximum power of 5 in 24![245]+[2452]+…..= 4 + 0= 4[242]+[2422]+[2423]+[2424]+…..= 12 + 6 + 3 + 1= 22Maximum power of 5 in 24![245]+[2452]+…..= 4 + 0= 4

Number of 5’s in 24! is 4

And the numbers of 2’s are more than 4

So the maximum possible pairs of 2 and 5 that can be made are 4 so the number of zeros in 24! are 4

Now 24! also have power of 3! i.e. 6 . So the whole expression would have 4 x 6 = 24 zeros in the end.

### Extra Problems for ‘Number of Zeros’ Question Type

**Extra Problems for ‘Number of Zeros’ Question Type**

**Example 1: Find the number of zeros in 2 ^{145} x 5^{234 }. **

**Solution**:When we see the question it looks like a very difficult question but this type of question involving number of zeros is very simple and can be solved in seconds. Here the question is asking for the number of zeros, now we have to look for the pairs of 2x 5, so the maximum pairs we can form are 145 because we have the maximum power of 2 is 145 , so the number of zeros in this case are 145.

**Example 2**: **How many numbers of zeros are there in following expression**

2^{11} x 125^{3} x 7^{11}+ 3^{11} x 7^{11}x 2^{10 }x 5^{1}+ 11^{12} x 2^{13} x 5^{125}**Solution**: To explain the solution we must know something about the nature of zeros, before doing this lets have a simple example, the number of zeros in the end of

10+100+1000+……………..+ 10000000000000000000 are 1 because we can take 10 common out of this expression and write it as 10(1+10+100+……………..+ 1000000000000000000) And clearly the term present inside the bracket has unit digit 1 which when multiplied with 10 results into 1 zero at its end.Thus number of zeros in any such expression will depend on the least zero holder in the expression.

Now using the same logic we can say least number of zeros will be contributed by second term i.e. 3^{11} x 7^{11}x 2^{10 }x 5^{1} as it contains only single power of 5 .Hence it contains only 1 zero, so the number of zeros for whole expression is 1 .

**Example 3**:**Find the number of zeros at end of ****5 x 10 x 15 x 20 x 25 x 30 x 35…………………………………… x 240 x 245 x 250 ****Solution**: We can take 5 common out of this expression and write it as

5^{50} (**1 x 2 x 3 x 4…………………………………… x 49 x 50)**

5^{50} (**50!)**

To find number of zeros we first find

Maximum power of 2 in 50!

[50/2]+[50/2^2]+[50/2^3]+[50/2^4]+[50/2^5]+…..= 25 + 12 + 6 + 3 +1= 47

Maximum power of 5 in 50![50/5]+[50/5^2]+[50/5^3]+…..= 10 + 2= 12

Thus maximum power of 5 present in given expression is 62 and maximum power of 2 present in given expression is 47. Hence number of zeros will be 47.

**Assignment: **__Questions:__

1 : Find the number of zeros in 75!

a) 16

b) 18

c) 20

d) 21

Ans: b

Solution:

Maximum power of 5 in 75!

[75/5]+[75/5^2]+[75/5^3]+…..= 15 + 3= 18

Hence number of zeros will be 18.

2: Find the number of zeros in 255!

a) 63

b) 52

c) 62

d) 65

Ans: a

Solution:

Maximum power of 5 in 255!

[255/5]+[255/5^2]+[255/5^3]+[255/5^4]+…..= 51 + 10 + 2= 63

Hence number of zeros will be 63.

3: Find the number of zeros in 135!

a) 25

b) 30

c) 33

d) 32

Ans: c

Solution:

Maximum power of 5 in 135!

= 27 + 5 + 1

= 33

[135/5]+[135/5^2]+[135/5^3]+…..= 27 + 5 + 1= 33

Hence number of zeros will be 33.

4: Find the number of zeros in n! where n is number between 66 to 69

a) 12

b) 14

c) 15

d) 16

Ans: c

We pick any value of n between 66 and 69. Number of zeros will be same for any value we pick between 66 and 69 say 68

Maximum power of 5 in 68!

= 13 + 2

= 15

[685]+[6852]+[6853]+…..= 13 + 2= 15">[685]+[6852]+[6853]+…..= 13 + 2= 15[685]+[6852]+[6853]+…..= 13 + 2= 15**Hence number of zeros will be 15.**

5: Find the number of zeros in 350!

a) 84

b) 85

c) 86

d) 87

Ans: c

Solution:

Maximum power of 5 in 350!

= 70 + 14 + 2

= 86

[350/5]+[350/5^2]+[350/5^3]+[350/5^4]+…..= 70 + 14 + 2= 86* So the number of zeros in the end of the 350! are 86*.

### Remainders (Part-1)

**Remainders: Part-1**

When a particular number is divided by any another number(divisor) we get quotient and remainder.

For example: On dividing 9 with 4 we get 2 as the quotient and 1 as the remainder. Here 9 is dividend and 4 is divisor.

**Tool tip:**The value of the remainder is always less than the divisor.**Find the remainder when 39 is divided by 6 ?**

We can write 39 = 6(6) + 3. Here quotient is 6 and 3 is remainder. Note that remainder 3 is less than divisor 6.

**Find the remainder when 79 is divided by 11 ?**

We can write 79 = 11(7) + 2. Here quotient is 7 and 2 is remainder. Note that remainder 2 is less than divisor 7.

**If divisor is same then it hardly creates any difference whether we add numbers and then divide by divisor to calculate remainder or we separately calculate remainders and then add.**

Let’s understand first of all whether we deal numbers separately while calculating remainders or we deal together it doesn’t create any difference. Lets understand meaning of above with help of example

Suppose we have 53 sticks in a bundle and we are told to divide it in groups of 5. So we all know that while dividing 53 in groups of 5, three sticks will be left.

Let’s consider another bundle of 21 sticks and we are told to divide it in groups of 5. So we all know that while dividing 21 in groups of 5, one stick will be left.

Now if we add the above two bundles we have total of 53 + 21=74 and again divide it in groups of 5

Then we will be left with 4 sticks.

Hence instead of mixing we can treat them by two groups as well and we can make groups of 5 separately and total numbers of remaining sticks are simply 4(3+1)i.e. addition of remaining sticks in the two groups separately.

Since 4 is lesser than 5 we cannot make another group of 5 from remaining sticks

But if instead of 21 sticks we had 22 sticks then the remainder would be 2 and the total remaining sticks would have been 5, and then we can make a new group of 5 with the remaining sticks with no remnant left.

### Remainders (Part-2)

**Remainders: Part-2****Type: Difference between divisor and remainder is same****On dividing by p, q and r the number gives different remainders a, b and c where ****p-a=q-b=r-c**

Let’s take an example to understand this

__Example__**What is the minimum two-digit number which when divided by 3, 4, 5 leaves 1, 2, 3 as the remainder respectively?****Solution:**

Let P be the required two digit number .As given in the question

P = 3 d + 1 =3 (d + 1) -2

P = 4 e + 2= 4( e + 1) -2

P = 5 f + 3= 5 (f+1) – 2

In each of above cases we observe difference between divisor and remainder is same.

Hence we have same remainder -2 when divided by 3,4,5

Thus number N must be in the form L.C.M. (3,4,5) k – 2 =60 k – 2

Since we are looking for two digit number putting k =1 we get 58 as minimum two-digit number which when divided by 3, 4, 5 leaves 1, 2, 3 as the remainder respectively.

__Example__**What is the minimum three-digit number which when divided by 2, 5, 6 leaves 1, 4, 5 as the remainder respectively?****Solution:**

Let P be the required three digit number .As given in the question

P = 2 d + 1 =2 (d + 1) -1

P = 5 e + 4 = 5( e + 1) -1

P = 6 f + 5 = 6 (f+1) – 1

In each of above cases we observe difference between divisor and remainder is same.

Hence we have same remainder -1 when divided by 2,5,6

Thus number N must be in the form L.C.M. (2,5,6) k – 1 =30 k – 1

Since we are looking for two digit number putting k =34 we get 1019 as minimum three-digit number which when divided by 2, 5, 6 leaves 1, 4, 5 as the remainder respectively.

*Assignment: ***QUESTIONS:****1.** What is the remainder when 72 + 51 is divided by 8 ?

a)1

b)2

c)3

d)5

Answer – c**Solution : **Since 72 is multiple of 8 hence it leaves remainder zero and 51 leaves remainder 3 when divided by 8.Hence 72 + 51 leaves remainder 3 when divided by 8.

**2.** What is the remainder when 111 + 68 + 48 is divided by 11 ?

a)5

b)7

c)9

d)None

answer – b**Solution:**

111 leaves remainder 1 when divided by 11

68 leaves remainder 2 when divided by 11

48 leaves remainder 4 when divided by 11

Thus 111 + 68 + 48 leaves remainder 1+2+4 =7 when divided by 11**3.** What is the remainder when 42×56 is divided by 8 ?

a)0

b)1

c)2

d)3

answer – a**Solution :**

56 is a multiple of 8 hence remainder is zero.

**4.**What is the remainder when 48×65 is divided by 7 ?

a)1

b)2

c)3

d)5

Answer – 5

48 leaves remainder 6 and 65 leaves remainder 2 so 48 65 divided by 7 gives 6×2=12 but 12 is greater than 7 hence we further divide by 7 and we obtain 5 as remainder.

**5.** What is the remainder of 84x71x41 by 9?

a)2

b)3

c)6

d)8

answer-b

84 leaves remainder 3, 71 leaves remainder 8 and 41 leaves remainder 5 when divided by 9. So 84x71x42 divided by 9 gives 3 8 =120 but 120 is greater than 9 hence we further divide by 9 and we obtain 3 as remainder.

### Remainders (Part-3)

**Remainders: Part-3****Problem Type 1: What will be the remainder when p + q +r +… is divided by d**

__Example__**What will be the remainder when 63 + 67 +81 is divided by 11 ?****Solution**: Instead we add up all numbers, lets do it separately

63 when divided by 11 gives 5 as quotient and 8 as remainder.

67 when divided by 11 gives 6 as quotient and 1 as remainder.

81 when divided by 11 gives 7 as quotient and 4 as remainder.

Now the remainders are 8, 1 and 4 respectively.

By adding all the remainders 8+1+4=13, the sum of remainders is greater than the divisor

Therefore again the sum of remainders is divided by the divisor i.e. by 11

On dividing 13 by 11,quotient will be 1 and 2 will be the remainder

Hence the remainder on dividing 63 +67 +81 by 11 will be 2.

**Tooltip :**

Now the question arises that by dividing 53 sticks in groups of 5 we are left with 3 extra sticks. Then how many more sticks should be added to this bundle so that the left out sticks are again 3?

Answer

If we want left out sticks to be 3 again then we should add minimum 5 more sticks or

10,15,20 ,….., 5n.

But remember the additional sticks should be multiple of 5.

53 when divided by 5 gives 10 as quotient and 3 as remainder and adding multiple of 5 is equivalent to adding remainder zero every time.

Let’stake an example to understand this

**Problem Type 2:When the remainder is same on dividing a number ‘a’ by different divisors.**__Example__

What is the minimum two-digit number which when divided by 3, 4 and 5 leaves 2 as the remainder each time?

**Solution:**

The question would have been easy if we were asked about minimum number, which is completely divisible by three numbers 3,4 and 5 then we can say it should be least common multiple of 3,4 and 5 i.e. 60. Again 60 is the number which is completely divisible by 3,4,5 but if we want 2 remainder in each case then add 2 in 60 i.e. 62 will be the minimum two digit number which when divided by 3, 4 and 5 leaves 2 as the remainder each time.

### Remainders (Part-4)

**Remainders: Part-4**

In this article, for calculating remainders we will use concept of cyclicity.

Let’s take an example of questions based on concept of remainders.

__Example__**Find the remainder when 47 ^{123} is divided by 7?**

**Solution:**

We will do this question step by step

Step 1: We divide 47 with 7 to get remainder as 5.

Step 2: Here in the question 47 is multiplied by itself 123 times. Now understand every time when we will divide 47 with 7 the remainder will be 5.

Step 3: Hence we can say finding remainder of 47

^{123}is equivalent to finding remainder of 5

^{123}when divided by 7.

So we start observing powers of 5.

5

^{2}gives remainder 4.

5

^{3}gives remainder 6.

5

^{4}gives remainder 2.

5

^{5}gives remainder 3.

5

^{6}gives remainder 1.

Hence after 6th power remainders will start repeating. So we have a cycle of 6 and thus we divide 123 by 6

i.e. 123= 6(20) + 3

Hence 5 ^{123}

-> 5 ^{6(20) +3}

-> 5 ^{6(20) }.5 ^{3 }-> 1.6 =6 (As 5^{6} gives remainder 1 and 5^{3} gives remainder 6)

Hence the remainder of 47^{123} when divided by 7 is 6.

__Example__**Find the remainder when 79 ^{644} is divided by 9 ?**

**Solution:**

We will do this question step by step

Step 1: We divide 79 with 9 to get remainder as 7.

Step 2: Here in the question 79 is multiplied by itself 644 times. Now understand every time when we will divide 79 with 9 the remainder will be 7.

Step 3: Hence we can say finding remainder of 79

^{644}is equivalent to finding remainder of 7

^{644}when divided by 9.

So we start observing powers of 7.

7

^{2}gives remainder 4.

7

^{3}gives remainder 1.

Hence after 3rd power remainders will start repeating. So we have a cycle of 3 and thus we divide 644 by 3 i.e. 644= 3(214) + 2

Hence 7

^{644 }-> 7

^{3(214) + 2 }-> 7

^{3(214) }.7

^{2}-> 1.4 = 4 (As 7

^{3}gives remainder 1 and 7

^{2}gives remainder 4)

Hence the remainder of 79

^{644}when divided by 9 is 4.

*Assignment:***QUESTIONS**

**1.** Find the remainder when 2^{57} is divided by 3?

a)0

b)1

c)2

d) None of these**Solution:**

Answer – c

So we start observing powers of 2.

2^{1} gives remainder 2.

2^{2} gives remainder 1.

2^{3} gives remainder 2.

2^{4} gives remainder 1.

Hence we can observe odd power of 2 gives remainder 2 and even power gives remainder 1.

Hence 2^{57} will give remainder 2 when divided by 3.

**2**. Find the remainder when 3^{91} is divided by 5 ?

a)1

b)2

c)3

d)4**Solution:**

Answer-b

So we start observing powers of 3.

3^{1} gives remainder 3.

3^{2} gives remainder 4.

3^{3} gives remainder 2.

3^{4} gives remainder 1.

Hence after 4th power remainders will start repeating. So we have a cycle of 4 and thus we divide 91 by 4 and 3^{91} will give same remainder as 3^{3} i.e. 2.

Hence the remainder of 3^{91} when divided by 5 is 2.

**3.** What is the remainder when 15^{48} is divided by 4?

a)1

b)2

c)3

d)0**Solution :**

Answer – b

Finding remainder of 15^{48} is equivalent to finding remainder of 3^{48} when divided by 4.

So we start observing powers of 3.

3^{1} gives remainder 3.

3^{2} gives remainder 1.

3^{3} gives remainder 3.

3^{4} gives remainder 1.

Hence we can observe odd power of 3 gives remainder 3 and even power gives remainder 1.

Hence 3^{48} will give remainder 1 when divided by 4.

Hence the remainder of 3^{48} when divided by 4 is 1.

**4.** Find the remainder when 13^{48}is divided by 9?

a)1

b)2

c)3

d)4**Solution :**

Answer- a

Finding remainder of 13^{48} is equivalent to finding remainder of 4^{48} when divided by 9.

So we start observing powers of 4.

4^{1} gives remainder 4.

4^{2} gives remainder 7.

4^{3} gives remainder 1.

Hence after 3rd power remainders will start repeating. So we have a cycle of 3 and we know 48 is divisible by 3 so answer will same as remainder of 4^{3}.

Hence the remainder of 13^{48} when divided by 9 is 1.

**5.** What is the remainder when 28^{71} is divided by 7?

a)1

b)2

c)3

d)0**Solution :**

answer – d

Since 7 is a factor of 28 hence remainder will be zero.

#### Remainders (Part-5)

**Remainders: Part-5**

In this article we will discuss calculating remainders that involve hefty mathematical expressions. We will discuss each and every mathematical expression one by one.

**Suppose we have two numbers “a” and “b”. If “a” is in form of 7n +4 and “b” is in form of 7m+1.**

**Addition****Find remainder when a+b is divided by 7?**

Given a = 7n +4

and b = 7m + 1

So when we add them we will have a +b = 7n + 7m + 4 + 1 = 7 (n + m) + 5

That means we have a multiple of 7 and a remainder 5, because five is smaller than the 7 so it cannot be divided further and hence this 5 is our remainder.

**Difference****What will be the remainder when a – b is divided by 7 ?**

Given a = 7n +4

and b = 7m + 1

So when we subtract them we will have a – b= (7n +4) – (7m + 1)

= 7(n – m ) + 4 – 1

= 7(n – m ) + 3.

That means we will have a multiple of 7 with addition of 3 and because three

is smaller than the 7 so it cannot be divided further and hence this 3 is our remainder .** ****Multiplication****What will be the remainder when a x b is divided by 7 ?**

Given: a = 7n +4

and b = 7m + 1

So when we multiply them we will have a x b = (7n +4) x (7m + 1)

= 7n(7m + 1) + 4(7m + 1)

= 49nm + 7n + 28nm + 4. Here we can see that each and every number is a multiple of 7 except 4. Hence this 4 is our remainder.

Let’s see application of above in problems

__Example__**What will be the remainder when ****(576 + 453 -34 – 566 +6734 x 234 x 345 x 4564) is divided by 8 ?****Solution:**

So to find the remainder we will not solve all the expression we will divide it one by one separately and apply same operations on remainders as we have in the numbers.

First divide 576 by 8 and remainder is 0

Then divide 453 by 8 and remainder is 5

Then divide 34 by 8 and remainder is 2

Then divide 566 by 8 and remainder is 6

Then divide 6734 by 8 and remainder is 6

Then divide 234 by 6 and remainder is 2

Then divide 345 by 8 and remainder is 1

Then divide 4564 by 8 and remainder is 4

Now apply same operations between remainders and solve

0 + 5 -2 -6 +6 x 2 x 1 x 4

= 45

Now 45 is greater than the 8 so we will again divide the result with 8.

So divide 45 by 8 gives remainder 5 so the remainder of the whole expression is 5.