Factor Calculator

A factor (divisor) of a positive integer n is any whole number that divides n evenly with no remainder. The number 1 and n itself are always factors — every integer has at least these two (unless n = 1, which has only one factor). To find all factors efficiently, test divisors from 1 up to √n. If i divides n, both i and n/i are factors. This pairing method finds all divisors in O(√n) time instead of checking every number up to n.

Example: factors of 36. Test 1: 36/1 = 36 → factors 1, 36. Test 2: 36/2 = 18 → 2, 18. Test 3: 12 → 3, 12. Test 4: 9 → 4, 9. Test 6: 6 → 6, 6 (perfect square, counted once). Test 5: no. Stop at √36 = 6. Complete list: 1, 2, 3, 4, 6, 9, 12, 18, 36 — nine factors total. Factor pairs multiply to n: (1,36), (2,18), (3,12), (4,9), (6,6).

Prime numbers have exactly two factors: 1 and themselves. The number 17 is prime because no integer between 1 and 17 divides it evenly. Composite numbers like 36 have more than two factors. The factor count and sum are useful in number theory: σ(n) denotes the sum of all divisors. Perfect numbers equal the sum of their proper divisors (excluding n itself) — 6 = 1 + 2 + 3 and 28 = 1 + 2 + 4 + 7 + 14 are the smallest examples.

Factors connect to GCF and LCM via the GCF calculator and fraction simplification via the fraction calculator. Prime factorization (see the prime factorization calculator) decomposes n into prime building blocks; the factor list combines those primes in all possible products. For large n, factor count grows slowly compared to n itself — highly composite numbers like 360 have many factors, while primes have the minimum.

Applications include simplifying fractions (divide numerator and denominator by their GCF), solving divisibility puzzles, checking whether n is prime, finding common denominators, and enumerating rectangular grid dimensions with area n. Enter any positive integer up to the calculator's limit and receive the sorted factor list, paired representation, count, sum, and a prime/composite verdict instantly.

Divisors are the same as factors. Negative factors exist in abstract algebra but this tool focuses on positive divisors standard in elementary and middle school mathematics. Verify homework, explore number patterns, or confirm primality before using numbers in cryptographic contexts where true primality testing requires advanced algorithms.