Prime numbers don't factorize; they're already prime.
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23 x 3 x 5 = 120
No. For example: 4181 / 37 = 113 so it can't be prime. But 4181 is the first composite number in the Fibonacci sequence with a prime index. ;)
It's more efficient. You want to end up with all prime numbers. You could divide by a composite number, but you'd just have to break that number down later. It saves steps.
23 x 32
You can use prime numbers to factorize numbers, a very useful tool