What is the prime factorization of 180?

The prime factorization of 180 is 5 × 2 × 2 × 3 × 3. We can also represent this prime factorization as 5 × 2² × 3². We are going to use a factor tree to find the prime factorization of 180. If we are trying to find the prime factorization of a general number, x, the steps are as follows:

1. Let x be the top of the tree, and draw two branches coming out of x.
2. Find two numbers, other than 1 and x, with a product of x. Place these at the bottom of each of the branches.
3. If the numbers at the bottom of the branches are prime, then we can stop, and this is the prime factorization of x, but if either of the numbers are not prime, then we repeat these steps for that number.
4. Continue this until all of the numbers at the ends of the branches are prime. These prime numbers make up the prime factorization of x.

Let's create a factor tree for 180. We place 180 at the top of the tree, and draw two branches. Then we find two numbers that multiply to 180. There are a number of different options here, but all of them will result in the same prime factorization, so it doesn't matter which ones we decide to use. Let's start with 10 and 18. Since neither of these are prime, we need to repeat these steps for each of the numbers, and continue to do so until all the numbers at the end of the branches are prime. This whole process is shown in the image.

We see that the numbers at the end of the branches are 5, 2, 2, 3, and 3. Therefore, the prime factorization of 180 is 5 × 2 × 2 × 3 × 3, or 5 × 2² × 3².

Please see the attached file for the complete solution