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.

Prime factorization - Definition

The prime factorization of a number is the way of expressing the number as a product of prime numbers. The prime factorization is written in only way as the order of factors does not matter.

Prime factorization method

The method for finding the prime factors involves divisibility tests and trial divisions. The quotient obtained in each level of division is again divided by a prime factor. The division process continues till a prime quotient is got. The Product of all the prime divisors and the final prime quotient gives the prime factorization for the number.

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 = 23 x 32.
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

Prime factorization for algebraic terms can also be written as it is done for integers.
Consider the expression 8x2y3. This can be written in factored form as
8x2y3 = 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
8x2y3 = 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 16a4b2  and 24a3b3.
Writing the prime factorization of the two expressions,
16a4b2 = 2 . 2 . 2 . 2 . a.a.a.a.b.b                                               and          24a3b3 = 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 = 8a3b2.