**PRIME FACTORIZATION**

All composite numbers can be broken up into their factors.

When the factors of a number are all prime numbers, it is called the prime factorisation of the number.

There are two ways of finding the prime factors of a number.

**(a) Prime Factorization Method**

56 = 2×28

= 2×2×14

= 2×2×2×7

The prime factors of 56 = 2×2×2×7.

**(b) Division Method**

The prime factors of 156 = 2×2×3×13

**EVEN AND ODD NUMBERS**

A numbers that is a multiple of 2 is called an even number. All even number and with the digits 0, 2, 4, 6, or 8.

A number that is not a multiple of 2 is called and odd number. All odd numbers end with the digits 1, 3, 5, 7, or 9.

The product of an even number and any other number is always even. The product of two odd numbers is always odd.

**MORE DIVISIBILITY RULES**

4 If the number formed by its last digits is divisible by 4 or are Zeroes.For example, 2832, 916, 83001 6 If the number is divisible by both 2 and 3.

For example, 384, 1602, 2982 8 If the number formed by its last three digits is divisible by 8.

For example, 7112, 13760, 26584 11 If the difference between the sum of digits in odd places and sum of digits in the even places is 0 or divisible by 11.

For example, 9845

9 8 4 5

↓ ↓ ↓ ↓

Odd place Event place Odd place Event place

9+4 = 13

8+5 = 13

Difference = 0

**HIGHEST COMMON FACTOR**

Let us look at the common factors of 12 and 18.

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

Factors of 18: 1, 2, 3, 6, 9, 18

The common factors of 12 and 18 are 1, 2, 3, and 6

The Highest Common Factor (HCF) of 12 and 18 is 6.

*The HCF of two numbers is the greatest number that can divide both numbers without leaving any remainder.*

So, in the above example, 6 is the greatest number that can divide both 12 and 18 without leaving a reminder.

**CHECKPOINT**

The HCF of two or more numbers is always smaller than or equal to the numbers.

There are three methods used to find the HCF of numbers.

### 1. Prime Factorization Method

Find the HCF of 24 and 32 using the prime factorization method.

2×2×2 = 8

HCH of 24 and 32 is 8.

**Prime Factors of 24**

2 ** **2

2 2

**Prime Factors of 32**

2 2

2 2 2

**Common prime Factors of 24 and 32**

HCF of 24 and 32 = 2×2×2 = 8