Prime Factorization: Divide, Record, and Multiply Back

Break the number into prime building blocks

A prime number is greater than 1 and has exactly two positive divisors: 1 and itself. Prime factorization expresses an integer greater than 1 as a product of primes. The order of those primes does not change the product.

For 360, divide repeatedly: 360 ÷ 2 = 180; 180 ÷ 2 = 90; 90 ÷ 2 = 45; 45 ÷ 3 = 15; 15 ÷ 3 = 5; 5 ÷ 5 = 1. The recorded divisors give 360 = 2³ × 3² × 5.

Use the exponents to count divisors

A factor of 360 can contain zero through three copies of 2, zero through two copies of 3, and zero or one copy of 5. Those independent choices give (3 + 1)(2 + 1)(1 + 1) = 24 positive divisors.

The distinct prime factors are only 2, 3 and 5. That short list is not the complete factor list. For example, 12 is a factor because 12 × 30 = 360, but 12 is not prime.

Handle one correctly

One is neither prime nor composite. It is a factor of every positive integer, but it is not included as a prime factor. Otherwise extra copies of 1 could be inserted indefinitely without changing any product.

After factoring, always multiply back. If an exponent is missing, that check reveals the error. The number dictionary shows each repeated-division step so you can verify both the factors and their multiplicities.

Check the mathematical definitions

For fractions, percentages and worked arithmetic, see OpenStax Prealgebra. For number definitions, see OpenStax on prime and composite numbers and Wolfram MathWorld on digital roots. Worked examples on this page are calculated for this guide.