Break positive integers into prime products such as 360 = 2³ × 3² × 5.
Overview
Every integer greater than 1 factors uniquely into primes (fundamental theorem of arithmetic). Prime factorization powers GCF and LCM via highest and lowest exponents per prime.
Trial division by primes 2,3,5,7,... is efficient for classroom sizes. Larger numbers use optimized algorithms, but the result structure is the same: primes with exponents.
Factor trees visualize splitting composites until all leaves are prime. The product of leaf primes equals the original number regardless of split order.
Use factorization to simplify radicals √(72) = 6√2, solve divisibility puzzles, and understand why 1 is not prime (it would break unique factorization).
Steps
Provide n > 1.
Start with 2, then 3, 5, ...
Count repeats for each prime.
Multiply factors to reproduce n.
Combine exponents across numbers as needed.
Copy 2^a × 3^b × ... notation.
Math
n = p1^a1 × p2^a2 × ... with primes pi
Repeated division by primes until quotient is 1 yields unique prime exponents for n>1.
Worked cases
360 = 2×2×2×3×3×5
2³ × 3² × 5.
No divisors besides 1 and 97
97 is prime.
128 = 2^7
2⁷.
Why use this
Questions