Prime Factorization — Mathematics Knowledge System

Overview

Prime factorization is the process of breaking a whole number down into the unique set of prime numbers that multiply together to produce it. A prime number is a whole number greater than 1 whose only factors are 1 and itself (2, 3, 5, 7, 11, ...). Every whole number greater than 1 has exactly one prime factorization (up to the order of the factors) — a fact so fundamental it's called the Fundamental Theorem of Arithmetic — and this decomposition is the key tool for finding GCFs, LCMs, and simplifying fractions systematically.

Intuition

Think of prime numbers as the 'atoms' of multiplication: every whole number greater than 1 is either itself prime, or can be split into a product of primes, just as every molecule is built from atoms. To factor 60, you keep dividing by the smallest prime that works: 60 = 2×30 = 2×2×15 = 2×2×3×5. No matter which primes you happen to pull out first, you always end up with the same multiset of prime factors.

Formal Definition

Definition

The Fundamental Theorem of Arithmetic states that every integer n > 1 can be written as a product of prime numbers, and this representation is unique up to the order of the factors.

n = p_1^{e_1} p_2^{e_2} \cdots p_k^{e_k}

p_1, \ldots, p_k distinct primes, e_i \geq 1

Prime factorization

Properties

Uniqueness

\text{Every } n > 1 \text{ has exactly one prime factorization, up to order.}

GCF via prime factorization

\gcd(a,b) = \prod_i p_i^{\min(e_i, f_i)}

Condition: where a = \prod p_i^{e_i}, b = \prod p_i^{f_i}

Worked Examples

Repeatedly divide by the smallest prime that divides evenly.
$60 = 2\times30 = 2\times2\times15 = 2\times2\times3\times5$

Answer: 60 = 2² × 3 × 5

Practice Problems

Difficulty 2/10

Find the prime factorization of 84.

Common Mistakes

Common Mistake

Stopping the factorization before every factor is prime, e.g. writing 60 = 4 × 15.

A prime factorization must break every composite factor down further until ALL factors are prime: 60 = 4 × 15 = (2×2) × (3×5) = 2² × 3 × 5.

Summary

Prime factorization writes a whole number as a product of prime numbers.
By the Fundamental Theorem of Arithmetic, this factorization is unique up to order.
Repeated division by the smallest possible prime is a reliable method for finding it.

References

WebsiteWikipedia — Integer factorization