What is Factorization?
Factorization means taking a number and figuring out what multiplication results in that number1. For example, the factorization of 6 is 2 times 3, because 2 times 3 equals 6.
In a multiplication such as 2 * 3 = 6, we call the numbers being multiplied, namely 2 and 3, the factors, while the result is called the product. Hence, factorization just means figure out which factors we need to multiply to get a given number as the product.
A factorization doesn't need to have only two factors. since 2 * 2 * 3 = 12, a valid factorization for 12 would be 2 * 2 * 3.
A number doesn't have a single factorization. Since 2 * 6 = 12 and 2 * 2 * 3 = 12, both would be valid factorizations for 12. We also aren't restricted to factoring toward smaller numbers. 6 = 12 * 0.5, for example, so "12 * 0.5" is a valid factorization for "6." Finally, we can have infinite factorizations for a number given the fact that 5 = 5 * 1, i.e. we can always add a "times one" factor to the end to have a distinct factorization. So 5 * 1, 5 * 1 * 1, and 5 * 1 * 1 * 1 are all valid factorizations for the number 5. Of course, this wouldn't be useful for anything, but it's still technically a factorization.
The smallest possible factorization for a natural number that consists only of natural numbers (positive, whole numbers) is called its prime factorization. For example, 2 * 6 is a factorization for 12, but we can further factorize 6 into 2 * 3. This means 2 * 6 is not a prime factorization. Meanwhile, we can't factorize 2 * 2 * 3 into anything using only positive whole numbers but "2 * 1 * 2 * 1 * 3 * 1." Since that wouldn't make the factors smaller, that means "2 * 2 * 3" is the prime factorization for the number 12. Every natural number only has a single prime factorization.
A natural number that can't be factored into smaller factors is called a prime number. For example, since we can't factor 2 into anything but 2 * 1, that means 2 is a prime number. The same is true for 3. However we can factor 4 into 2 * 2, which are smaller factors, that means 4 is not a prime.
References
- https://mathworld.wolfram.com/Factorization.html, accessed 2026-03-19 ↩︎