LESSON: Prime Factorization

READ: Write the Prime Factorization Using a Factor Tree

Write the Prime Factorization of Given Numbers Using a Factor Tree

We can combine factoring and prime numbers together too. This is called prime factorization. When we factored numbers before, we broke down the numbers into two factors. These factors may have been prime numbers and they may have been composite numbers. It all depended on the number that we started with.

Example

Factor 36

36 can factor several different ways, but let’s say we factor it with 6 $\times$ 6.

These two factors are not prime factors. However, we can factor 6 and 6 again.

$6 &= 3 \times 2\\ 6 &= 3 \times 2$

3 and 2 are both prime numbers.

When we factor a number all the way to its prime factors, it is called prime factorization.

It is a little tricky to keep track of all of those numbers, so we can use a factor tree to organize. Let’s organize the prime factorization of 36 into a factor tree.

Notice that we write 36 as a product of its primes.

Is there any easier way to write this?

Yes, we can use exponents for repeated factors. If you don’t have any repeated factors, you just leave your answer alone.

$2 \times 2 &= 2^2\\ 3 \times 3 &= 3^2$

The prime factorization of 36 is $2^2 \times 3^2$ .