What Is Prime Factorization? A Beginner's Guide
Learn what prime factorization is, how to do it step by step, and why it helps with greatest common divisors, least common multiples, and simplifying fractions.
Prime factorization is an important idea in middle-school math. The name sounds intimidating, but all it means is breaking a number down into a product of prime numbers. In this guide we start with what a prime number is, then cover how to factorize and why it is useful.
What is a prime number?
A prime number is a whole number greater than 1 that can only be divided evenly by 1 and itself. The primes go 2, 3, 5, 7, 11, 13, and so on. The number 6, for example, is not prime because it can be divided by 2 and by 3.
One thing to remember: 1 is not a prime number. A prime is defined as having exactly two divisors — 1 and itself — and 1 has only one divisor.
How to factorize
The basic method is to divide by the smallest prime numbers in order. Let's factorize 36. Divide by 2 to get 18, then by 2 again to get 9. Since 9 is not divisible by 2, move to the next prime, 3: dividing gives 3, and dividing once more gives 1.
Listing the primes we divided by gives 2 × 2 × 3 × 3. Grouping repeated primes with exponents, we write 36 = 2^2 × 3^2. That is the prime factorization.
Why is it useful?
Prime factorization helps you find the greatest common divisor and least common multiple. Factorize two numbers, and the shared primes give the greatest common divisor, while collecting all the primes to their highest needed powers gives the least common multiple.
The same idea helps when simplifying fractions. Factorize the numerator and denominator, and any common prime factors show you what to cancel. Think of prime factorization as a tool that reveals the building blocks of a number.
Practice by doing
Prime factorization makes more sense once you try a few numbers yourself. Our prime factorization calculator shows the division steps and a prime check, making it ideal for checking your work and practicing.