Primes are the atoms of arithmetic, and factorization is the spectroscopy: 360 = 2³ × 3² × 5, uniquely, always. This calculator decomposes any number up to 10¹⁵, lists its factor pairs, and rules on primality.
The algorithm: patient division
Divide by 2 as many times as possible, then 3, then upward — any factor found is automatically prime, because its smaller factors were already stripped out. The search stops at √n: a number with no prime factor up to its square root is prime. That's why 97 needs only four checks (2, 3, 5, 7) to be certified.
What the exponents buy you
The compact form powers instant answers. Divisor count is Π(exponentᵢ + 1): 360 = 2³·3²·5¹ has 4×3×2 = 24 divisors — no listing required. Highest powers and lowest powers of shared primes give LCM and GCF respectively, the machinery behind the GCF & LCM calculator.
Factor pairs: the rectangle view
Every divisor d pairs with n/d — 36 gives 1×36, 2×18, 3×12, 4×9, 6×6. Geometrically these are the whole-number rectangles with area n, and the middle "square" pair (6×6) appears exactly when n is a perfect square — which is also why perfect squares have an odd divisor count.
Why the 10¹⁵ cap is honest
Trial division costs about √n steps — fine at 10¹⁵ (~30 million checks, milliseconds), hopeless at cryptographic sizes. That hardness isn't a bug of mathematics; it's the foundation: RSA encryption trusts that multiplying two 300-digit primes is easy but un-multiplying them is not. Related integer tools: number base converter and square root calculator.
Exact BigInt arithmetic; factor pairs listed up to 1,000,000.