Prime Factor

An important fact is that any number can be written as the product (multiplication) of prime numbers in one way. For example, 20 = 5 x 2 x 2 . This is the only way of writing 20 as the product of prime numbers. Writing a number in this way is called prime factor decomposition.
Example

Find the prime factor decomposition of 36.

We look at 36 and try to find numbers which we can divide it by. We can see that it divides by 2.
36 = 18 × 2

2 is a prime number, but 18 isn't. So we need to split 18 up into prime numbers. We can also divide 18 by 2.
18 = 9 × 2
and so 36 = 18 × 2 = 9 × 2 × 2

But we haven't finished, because 9 is not a prime number. We know that 9 divides by 3.
9 = 3 x 3.

Hence 36 = 9 × 2 × 2 = 3 × 3 × 2 × 2.

This is the answer, because both 2 and 3 are prime numbers.
Example

a and b are prime numbers, ab3 = 54. Find the values of a and b.

So ab3 is the prime factor decomposition of 54.
54 = 2 × 27 = 2 × 3 × 9 = 2 × 3 × 3 × 3 = 2 × 33
So a = 2 and b = 3.

Example 1