Prime factorization

Prime factorization is a way to break one large number into a product of smaller prime numbers. When you prime-factor a number, you end up with prime numbers whose product (when multiplied together) equals the original number.

For instance, the prime factorization of $12$ is $2\ast 3\ast 3$. Each factor is prime, and $2\ast 3\ast 3=12$.

Factor trees

A consistent way to find a prime factorization is to use a factor tree, as described in the chapter “GCF/LCM.” We’ll review the idea briefly here.

Start with the original number, then split it into two factors. Keep factoring each branch until every branch ends in a prime number. The following factor trees show the prime factorization of $16$ and $48$.

The prime factorization of $16$ is $2\ast 2\ast 2\ast 2$.

The prime factorization of $48$ is $3\ast 2\ast 2\ast 2\ast 2$.

Prime factorization is useful because the numbers at the ends of the branches are prime, so they can’t be factored any further. That means you’ve written the original number in its most broken-down form. This idea also shows up occasionally on the ACT.