A prime factor in Math is a factor of a number that simply happens to be prime.
With a prime number being a number that is only divisible exactly by 1 and itself.
Finding the prime factors of a non prime number, isn’t too dissimilar to attempting to find the factors of a number in general.
However we’re looking for factors that are prime numbers.
When it comes to the prime factorization of a number, we look to re-write a number as the product of prime factors multiplied together.
Prime Factorization of a Number Examples
(1.1)
We can look at a random number for an example, let’s say 56.
An effective approach is to keep dividing by prime numbers, until we get obtain an answer that is also prime number.
We start off by attempting to divide the number by the smallest prime number, which is 2.
1) 56 ÷ 2 = 28
2) 28 ÷ 2 = 14
3) 14 ÷ 2 = 7
We have obtained a prime number after three divisions.
If we multiply together the last prime number we get as an answer, with the prime numbers that were used to divide at each stage, this will result in the original number from the beginning.
With this case, it was just 2 that was used to divide at each stage.
But it was still used a total of three times before we obtained the prime number of 7.
2 × 2 × 2 × 7 = 8 × 7 = 56
This new sum with prime numbers can be shortened with an exponent/power, to give a neat the prime factor form for 56.
23 × 7 = 56
In this form, 2 and 7 are factors that are prime.
We have completed the prime factorization of 56.
(1.2)
We can try prime factorization with the number 45.
1) 45 ÷ 2 = 22.5 This doesn’t divide exactly, so look to divide by the next prime number instead, 3.
2) 45 ÷ 3 = 15
3) 15 ÷ 3 = 5 => 5 is a prime number.
3 was used at two stages of exact division before we obtained the prime number of 5 as a result.
3 × 3 × 5 = 9 × 5 = 45
This can be written as, 32 × 5 = 45.
(1.3)
This example looks at the prime factorization of a number that is larger, 238.
1) 238 ÷ 2 = 119
The next prime number after 2 that divides 119 exactly is 7.
2) 119 ÷ 7 = 17 => 17 is a prime number.
2 was used at one stage of exact division, as was 7, before the prime number of 17 was obtained as an answer.
2 × 7 × 17 = 238