Prime factorization is used to find prime factors of any given number. Prime numbers are numbers that only have factors of 1 and the number (like 3—the only factors of 3 are 1 and 3). Prime factorization allows you to easily find all the prime factors of a number, and all numbers have a prime factorization. There are several different ways to write prime factorization; we’re going to show you the “pine tree” method. No matter which way you write it, you’ll be getting the same numbers for your factorization.
In order to find the prime factors, you’re first going to look at the number you have and think of two numbers that multiply together to produce that number. Let’s look at the number 45. Can you think of two numbers that multiply together to get 45? Well, we know that 9 x 5 is 45. Now, we look at 9 and 5. Are either of these prime numbers? We know that 5 is a prime number because the only numbers that multiply to give us 5 are 1 x 5. We leave the 5 alone, and look at the 9. Is 9 a prime number? No, it’s not, so we need to think of what multiplies to give us 9. We know that 3 x 3 is 9, so 9 turns into 3 x 3. Now, we look at each 3. Is 3 a prime number? We know that both 3s are prime numbers, because the only thing that multiplies to give us 3 is 1 x 3. Thus, our prime factorization is 3 x 3 x 5.
We often use a specific way of writing this, as we mentioned earlier, called the “pine tree” format. Prime factorization does not have to be written this way, but it is easier to organize your problem. The “pine tree” format looks like this:
It may not look too much like a pine tree with a short factorization like 45 has. However, some numbers stretch out even more, and the shape looks similar to a pine tree. Each level of branches is a separate multiplication problem that includes the factors of the number. Remember, this is merely one way of writing the factorization. The main goal is for you to be left with the prime factors of whatever number you’re working with. Let’s look at one more example:
Each level of branches is another multiplication problem resulting in the previous number. From the top down, you have 144, which you can split into 12 x 12. Then, each twelve can be split into 3 x 4. Each 4 can be split into 2 x 2. Thus, the prime factorization (the numbers circled in blue in the diagram) is 2 x 2 x 2 x 2 x 3 x 3. Typically, we list the prime numbers in order from least to greatest. Since they’re being multiplied, you don’t have to keep them in factor pairs to order them.