5th Class Mathematics Factors and Multiples

Factors and Multiples

Introduction

We have studied about the operations on numbers. Now, we will study two important terms that is, 'factors' and 'multiples'. They are related to the operations of multiplication and division.

Factors

Factors of a number is the number, which divides the given number completely.

If a, b, c, d.... are factors of 'm' then 'm' will be exactly divisible by a, b, c, d....

How to Get Factors of a Number

To find all possible factors of a number, we have to find all the numbers, which divide the given number exactly.

Rules of Divisibility

1. The numbers which have 0, 2, 4, 6, or 8 at the unit place is divisible by 2. Ex: 5666, 5654, 130 are divisible by 2.
2. If sum of digits of a number is divisible by 3 then the number is divisible by 3. Ex: Sun- r the digits of 25441215\[=2+5+4+4+1+2+1=24\]. 24 is divisible by 3, therefore, 25441215 is divisible by 3.
3. If the number formed by its last two digits (ones and tens) is divisible by 4, the number is divisible by 4. Ex: 8928 is divisible by 4 as 28 is the last two digits which are divisible by 4.
4. If a number has the digit 0 or 5 at unit’s place, the number is divisible by 5. Ex: 5 is at the unit place in the number 5645, therefore, 5645 is divisible by 5.
5. If a number is divisible by 2 as well as by 3, the number is divisible by 6. Ex: the number 45822 is divisible by 6, since it is divisible by 2 as well as 3 as 2 is at unit's place and sum of the number is\[4+5+8+2+2=21\], which is divisible by 3.
6. If the number formed by its last three digits is divisible by 8, the number is divisible by 8. Ex: 2136 is divisible by 8. As the number formed by its last three digits is 136, which is divisible by 8.
7. If sum of digits of a number is divisible by 9, the number is divisible by 9. Ex: sum of digits of 78654588\[=7+8+6+5+4+5+8+8+3=54\] and 54 is divisible by 9. Thus 786545883 is divisible by 9.
8. If a number has the digit 0 at unit's place, the number is divisible by 10. Ex: 0 is at the unit place in the number 2549896980, 2549896980 is divisible by 10.

• Example:

Find all the possible factors of 15.

Solution: 1, 3, 5, 15 are factors of 15.

• Example:

Find all the possible factors of 56.

Solution: 1, 2, 4, 7, 8, 14, 28, 56 are factors of 56.

• Example:

Is 3 a factor of 4665366564?

Solution:

Yes.

Sum of digits of given number

 \[=4+6+6+5+3+6+6+5+6+4=51\] and 51 is divisible by 3.

Prime Number

The numbers which have only two factors, land the number itself are called prime numbers.

For example:

Factors of 5 = 1, 5

Factors of 7 = 1, 7

Factors of 11= 1, 11

Therefore, 5, 7, and 11 are prime numbers.

Twin Primes

Two consecutive prime numbers with the difference of 2 are called twin primes.

• Example:

Write two pairs of twin primes.

Solution:

(3, 5), (5, 7) as 3, 5 and 7 are prime numbers and the two pairs of numbers have a difference of 2.

Composite Number

A number which has more than two factors is called a composite number.

• Example:

Factors of 6 = 1, 2, 3, 6

Factors of 14 = 1, 2, 7, 14

Therefore, 6 and 14 are composite numbers

Perfect Number

If sum of all the factors of a number is twice of the number, the number is called a perfect number.

• Example:

6 is a perfect number because sum of factors of\[6=1+2+3+6=12\]   

Common Factors

The same factors of two or more than two different numbers are called common factors.

• Example:

Factors of 12 = 1, 2, 3, 4, 6, 12

Factors of 16 = 1, 2, 4, 8, 16

1, 2, 4 are the common factors of 12 and 16

Co-prime or Relatively Prime Numbers

If two numbers have only one common factor that is the numbers are called co-prime or relatively prime numbers.

• Example:

Factors of 8 = 1, 2, 4, 8

Factors of 9 = 1, 3, 9

1 is the only common factor of 8 and 9. Therefore, 8 and 9 are relatively co-prime numbers.

Multiples

When two or more than two numbers are multiplied with each other, the resulting number is the multiple of all that numbers. Like if\[A\times B=C\], C is multiple of both A and B. 

• Example:

Multiples of 7 = 7, 14, 21, 28, 35, .......

Multiples of 8 = 8, 16, 24, 32, 40, ......

Common Multiples

The same multiples of two or more than two different numbers are called common multiples.

• Example:

Multiples of 6 = 6, 12, 18, 24, 30, 36, 42, .....

Multiples of 4 = 4, 8, 12, 16, 20, 24, 28, 32, 36, .....

Common multiples of 6 and 4 = 12, 24, 36 etc.

• Example:

Multiples of 5 = 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, ....

Multiples of 6 = 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 72, ....

Common multiples of 5 and 6 = 30, 60, .....