Prime Factorization
Prime factorization writes a whole number greater than 1 as a product of prime numbers. Every integer has exactly one such factorization, apart from the order of factors.
Fundamental Theorem of Arithmetic
Every integer greater than 1 is either prime or can be uniquely expressed as a product of primes. This uniqueness (up to ordering) makes prime factorization a powerful tool.
For example, 60 = 2 × 2 × 3 × 5, often written as 2² × 3 × 5 using exponent notation.
How to Factor a Number
Start with the smallest prime (2) and divide repeatedly while divisible. Move to the next prime (3, 5, 7, …) until the quotient is 1.
For 84: divide by 2 → 42, divide by 2 → 21, divide by 3 → 7, divide by 7 → 1. Result: 84 = 2² × 3 × 7.
Applications
Prime factorization helps find the GCD and LCM of two numbers, determine whether a number is a perfect square or cube, and count divisors.
For GCD, take the lowest power of each shared prime. For LCM, take the highest power of each prime appearing in either factorization.
Examples
- 36 = 2² × 3²
- 100 = 2² × 5²
- Prime factorization of 17 is simply 17 (it is already prime).
FAQ
Can the order of prime factors change?
The order can change (2 × 3 × 5 = 5 × 2 × 3), but the set of primes and their exponents is unique. Order does not affect the value of the product.
How do you factor very large numbers?
Trial division works for small numbers. Large numbers use advanced algorithms and, in cryptography, factoring large composites is intentionally difficult to keep encrypted data secure.