Prime Factorization

Every positive integer greater than two is the product of a set of prime numbers known as a prime factorization. The prime factorization of 10 is 2 × 5 because 2 × 5 = 10, and 2 and 5 are both prime. The prime factorization of 5 is simply 5.

Finding prime factorizations

We can find the prime factorization of a positive integer if we keep breaking the integer down into a product of smaller integers until all we have left is a product of prime numbers.

To find the prime factorization of 24, we start by choosing two integers that multiply to 24, not counting 1 × 24.

We choose 2 × 12 = 24. Since 2 is a prime number, we can’t break it down any further. But 12 is a composite number, so we choose two integers that multiply to 12, not counting 1 × 12.

We choose 3 × 4 = 12. The number 3 is prime, but 4 is composite, so we choose two integers that multiply to 4, not counting 1 × 4.

We choose 2 × 2 = 4. Both numbers are prime. Multiplying all of the prime numbers together, we find that the prime factorization of 24 is 2 × 2 × 2 × 3.

Prime Factorizations Are Unique

An integer’s prime factorization is unique. No matter how we break down the integer, we will always find the same set of prime numbers when we are done.

For example, the prime factorization of 60 is always 2 × 2 × 3 × 5. We can test that by breaking down 60 by choosing different factors.

Practice Finding Prime Factorizations

Find the prime factorization of 60.

Start by tapping on a yellow circle to enter an integer that multiplies to 60.