A composite number can be factored in more than one way. Let us look at the different products that yield 24

24 = 1 x 24

24 = 2 x 12

24 = 3 x 8

24 = 4 x 6

24 = 2 x 3 x 6 and

24 = 2 x 2 x 3 x 3. Consider the factorization 24 = 2 x 2 x 2 x 3. Each factor in the product is a prime number.This is called the prime factorization for the number 24.

24 = 1 x 24

24 = 2 x 12

24 = 3 x 8

24 = 4 x 6

24 = 2 x 3 x 6 and

24 = 2 x 2 x 3 x 3. Consider the factorization 24 = 2 x 2 x 2 x 3. Each factor in the product is a prime number.This is called the prime factorization for the number 24.

## Prime factorization - Definition

## Prime factorization method

**Example:**

Find the prime factorization for 75.

The ones digit in the given number is 5. This means the number is divisible by 5. Division of 75 by 5 yields the quotient 15.

75 = 5 x 15. !5 is again divisible by both 3 and 5.The prime factorization of 15 = 3 x 5.

Hence the prime factorization of 75 = 3 x 5 x 5.

Note: If a prime factorization of a smaller composite factor is known, that can be used straight away without having to do the repeated division.

For example if we want to find the prime factorization of the number 72, we know it is divisible by 2 and

72 = 2 x 36. Suppose you already know the prime factorization of 36 as = 2 x 2 x 3 x 3, this can replace 36 in the product and the prime factorization of 72 is

72 = 2 x 2 x 2 x 3 x 3 or using the exponents, 72 = 2

^{3}x 3

^{2}.

This method is especially useful in writing the prime factorization of larger numbers as it is easier to factor bigger numbers into composite numbers.

## Prime Factorization in algebra

Consider the expression 8x

^{2}y

^{3}. This can be written in factored form as

8x

^{2}y

^{3}= 8.x.x.y.y.y Using the prime factorization of 8 as 8 =2 x 2 x 2, the prime factorization for the expression is

8x

^{2}y

^{3}= 2 . 2 . 2.x.x.y.y.y

This method is useful in finding the GCF of two or more algebraic expressions.

Find the GCF of the expressions 16a

^{4}b

^{2}and 24a

^{3}b

^{3}.

Writing the prime factorization of the two expressions,

16a

^{4}b

^{2}=

**2 . 2 . 2 .**2 .

**a.a.a.**a.

**b.b**and 24a

^{3}b

^{3}= 2 . 2 . 2 . 3 .

**a.a.a**.

**b.b**.b

Collecting the common factors the greatest common factor of the two expressions =

**2 . 2 . 2 . a.a.a.b.b = 8a**

^{3}b^{2}.