Prime Factorization Explained (with Examples)
Prime factorization breaks a number into the prime numbers that multiply to make it. Learn the method, how to count a number's divisors, and worked examples.
Try the prime factorization calculatorPut these numbers to workOpen →Prime factorization means writing a whole number as a product of prime numbers — numbers whose only divisors are 1 and themselves. Every integer greater than 1 has exactly one prime factorization (the fundamental theorem of arithmetic), which makes it the backbone of finding greatest common divisors, least common multiples, and simplifying fractions.
How to factor a number
- 1Divide by the smallest prime that goes in evenly — start with 2, then 3, 5, 7, and so on.
- 2Keep dividing the result by that same prime until it no longer divides evenly.
- 3Move to the next prime and repeat.
- 4Stop when the number left is 1. The primes you divided by, with their counts, are the factorization.
Worked example: 84
84 ÷ 2 = 42, ÷ 2 = 21. 21 is not even, so try 3: 21 ÷ 3 = 7. 7 is prime. So 84 = 2 × 2 × 3 × 7 = 2² × 3 × 7.
Counting the divisors
Once you have the factorization, add one to each exponent and multiply the results — that is how many divisors the number has. For 360 = 2³ × 3² × 5¹, it is (3 + 1)(2 + 1)(1 + 1) = 24 divisors. This trick saves you from listing them all by hand.
Why it is useful
- •Greatest common divisor (GCD): multiply the primes both numbers share, at the lowest power each appears.
- •Least common multiple (LCM): multiply every prime that appears in either number, at the highest power.
- •Simplifying fractions: cancel the prime factors the numerator and denominator have in common.
- •Checking if a number is prime: if it has no prime factors other than itself, it is prime.