Study Guide

Basic Operations - Prime Factorization

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Prime Factorization

Prime factorization is the process of breaking down a number into its prime factors.

Say what?

Okay, let's break that statement down:

  • Prime number: a number that's only divisible by 1 and itself.

    • 3 is only divisible by 1 and 3, so (spoiler alert!) it's prime.
    • 4, however, is divisible by 1, 2, and 4. It's not prime.
    • The first ten prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, and 29.
    • The number 1 doesn't technically count as a prime number because it only has one factor: itself.
  • Factor: a number that can be divided into another number. For example, 1, 2, 3, 4, 6, and 12 are all the factors of 12.

How to Do It

A factor tree is a good way to visualize a number's prime factors. It's kinda like a family tree but in reverse. We want to find a number's ancestors, a.k.a. all the prime numbers that we'd multiply together to get the number.

For example, what are the prime factors of 24? To make a factor tree, we start with the number 24 and draw two branches coming down from it. At the end of the branches we place any two factors of 24…except 24 and 1. Yeah, our tree doesn't like 1s. Let's use 8 and 3.

We next decide if either of those factors are prime and circle the factor if it is. We circle 3 and that branch ends right there. Now we keep factoring 8 by drawing two branches coming down from it, placing any two factors of 8, say 4 and 2. Then, we circle the 2 because it's prime.

We continue until all of the branches end in circles (that is, prime numbers).

And in a move of sheer brilliance, we write all of the circled numbers (prime factors) being multiplied, in order from smallest to biggest.

24 = 2 × 2 × 2 × 3

By the way, we can check our answer by multiplying all the prime factors together.

2 × 2 × 2 × 3 =
4 × 2 × 3 =
8 × 3 =

Sure enough, all our prime factors bring us back to 24, our original number, when we multiply 'em together.

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