Prime Factorization
Breaking any number into the prime building blocks that only it has.
The explanation
A prime number has exactly two factors: 1 and itself. 2, 3, 5, 7, 11 and so on. Everything else can be broken down into primes multiplied together.
To break a number down, keep splitting it until only primes are left. For 60: 60 = 6 × 10, then 6 = 2 × 3 and 10 = 2 × 5. So 60 = 2 × 2 × 3 × 5.
The useful part is that it does not matter how you start splitting. Begin with 60 = 4 × 15 instead and you land on the same four primes. Every number has one and only one prime recipe.
The Fundamental Theorem of Arithmetic states that every integer n > 1 factors into primes uniquely up to the order of the factors. Written in canonical form:
n = p₁^a₁ · p₂^a₂ · … · p_k^a_k
where the pᵢ are distinct primes in increasing order and each exponent aᵢ ≥ 1. For 60 this is 2²·3·5.
This canonical form turns several messy questions into bookkeeping. The number of factors of n is (a₁+1)(a₂+1)…(a_k+1), because each factor chooses an exponent from 0 up to aᵢ independently — 60 therefore has (2+1)(1+1)(1+1) = 12 factors. GCF and LCM also fall straight out of the exponents, which is the reason prime factorization is taught before fractions rather than after.
Worked example
Find the prime factorization of 84.
- 84 is even: 84 = 2 × 42.
- 42 is even: 42 = 2 × 21.
- 21 = 3 × 7, both prime.
Answer: 84 = 2² × 3 × 7
Common mistakes
- Calling 1 prime. It has only one factor, so it is neither prime nor composite.
- Stopping too early and leaving a composite number like 9 or 15 in the answer.