CALTRX

Factor Calculator

Find all factors and factor pairs of a given integer. Get instant results with step-by-step explanations.

Enter your values above to see the results.

Tips & Notes

  • ✓Only test divisors up to √n. Each factor below √n pairs with one above it — cuts work in half.
  • ✓Factor count = (a+1)(b+1)... for n=pᵃqᵇ. Faster than listing all factors.
  • ✓Perfect number: sum of proper factors = the number itself. 28 = 1+2+4+7+14 ✓.
  • ✓Abundant number: sum of factors > n. Deficient: sum < n. Most numbers are deficient.
  • ✓1 and n are always factors of n. Prime numbers have exactly 2 factors: 1 and themselves.

Common Mistakes

  • ✗Forgetting to test 1 and n — they are always factors of every positive integer.
  • ✗Testing only up to n/2 instead of √n — misses efficiency. √n is sufficient.
  • ✗Confusing factors (smaller, divide evenly) with multiples (larger, n divides them).
  • ✗In prime factorization, stopping at composite factors. 12=4×3 is incomplete — 4=2², so 12=2²×3.
  • ✗Factor count formula: (a+1)(b+1) not (a)(b). Each +1 accounts for the zero-power term.

Factor Calculator Overview

A factor of an integer n is any integer that divides n exactly with no remainder. Finding all factors of a number reveals its complete divisibility structure — how many ways it can be evenly split, what sizes of equal groups it can form, and what its prime building blocks are. This calculation underlies fraction simplification, GCF computation, divisibility tests, and the study of perfect numbers.

Finding all factors — systematic method:

Test every integer from 1 to √n. If k divides n evenly, both k and n÷k are factors.
EX: Factors of 36 → test 1 to √36=6: 1×36, 2×18, 3×12, 4×9, 6×6 → Factors: {1, 2, 3, 4, 6, 9, 12, 18, 36} — 9 total factors
Number of factors formula from prime factorization:
If n = p^a × q^b × r^c then total factors = (a+1)(b+1)(c+1)
EX: 360 = 2³ × 3² × 5¹ → factors = (3+1)(2+1)(1+1) = 4×3×2 = 24 total factors — without listing all 24 individually
Sum of factors formula (for perfect number testing):
EX: Sum of proper factors of 28 = (1+2+4+7+14) = 28 → 28 equals sum of its proper factors → 28 is a perfect number ✓
Factor analysis has deep consequences beyond simplification. The number 12 has an unusually large number of factors (1, 2, 3, 4, 6, 12) for its size — 6 factors total. This is why historical measurement systems adopted 12: 12 inches per foot, 12 months per year, 24 hours per day, 60 minutes per hour. These highly composite numbers divide evenly into many fractions, making practical arithmetic simpler before calculators existed. This gives (3+1)(2+1) = 12 factors — listing them systematically rather than by trial and error. Perfect numbers, where the sum of proper factors equals the number itself, have fascinated mathematicians since antiquity. Six: 1+2+3=6 ✓. Twenty-eight: 1+2+4+7+14=28 ✓. Euclid proved that if 2ⁿ−1 is prime (a Mersenne prime), then 2ⁿ⁻¹×(2ⁿ−1) is a perfect number — connecting prime numbers directly to perfect numbers. Only 51 perfect numbers are known. Whether any odd perfect number exists remains one of the oldest unsolved problems in mathematics.

Frequently Asked Questions

A factor of n is any integer that divides n with zero remainder. To find all factors: test every integer from 1 to √n. If i divides n, both i and n/i are factors. Example: factors of 36. Test 1: 36÷1=36 → factors 1 and 36. Test 2: 36÷2=18 → 2 and 18. Test 3: 36÷3=12 → 3 and 12. Test 4: 36÷4=9 → 4 and 9. Test 5: 36÷5=7.2 (not integer, skip). Test 6: 36÷6=6 → factor 6 (paired with itself). All factors: 1, 2, 3, 4, 6, 9, 12, 18, 36.

The number of factors = product of (each exponent + 1) from the prime factorization. 36 = 2²×3² → factors = (2+1)(2+1) = 9. Verify: 1, 2, 3, 4, 6, 9, 12, 18, 36 — exactly 9 ✓. Numbers with an odd count of factors are perfect squares (because the square root pairs with itself). 36 has 9 factors (odd) and √36 = 6 is an integer ✓. A prime number has exactly 2 factors: 1 and itself. Numbers with more than 2 factors are composite.

A prime number has exactly two factors: 1 and itself. 17 is prime (factors: 1, 17 only). A composite number has more than two factors. 15 is composite (factors: 1, 3, 5, 15). The number 1 is neither prime nor composite — it has only one factor (itself). Prime numbers are the building blocks of all integers through prime factorization. Every composite number can be expressed as a unique product of primes — a fact called the Fundamental Theorem of Arithmetic.

Factor pairs are pairs (a, b) where a × b = n. Factor pairs of 24: (1,24), (2,12), (3,8), (4,6). These pairs are always symmetric around √n. Pairs with both factors ≤ √n are in the lower half; pairs with both ≥ √n in the upper half; the middle pair straddles √n. Factoring into pairs is essential for binomial factoring in algebra: x² + 5x + 6 needs two numbers that multiply to 6 and add to 5 → (2,3) → (x+2)(x+3).

A perfect number equals the sum of its proper factors (all factors except itself). 6: proper factors 1+2+3 = 6 ✓. 28: proper factors 1+2+4+7+14 = 28 ✓. The next perfect numbers are 496 and 8128. Only 51 perfect numbers are known as of 2024, all even. Whether odd perfect numbers exist is one of mathematics' oldest unsolved problems. Perfect numbers are connected to Mersenne primes through the formula 2^(p−1) × (2^p − 1) when 2^p − 1 is prime.

In algebra, factoring polynomials mirrors integer factorization. Factor x² − 5x + 6: find two integers multiplying to 6 and adding to −5 → (−2)(−3) → (x−2)(x−3). Factor x² − 9: difference of squares = (x−3)(x+3). Factor x³ − 8: difference of cubes = (x−2)(x²+2x+4). Knowing integer factors of the constant term helps identify the rational roots of polynomials via the Rational Root Theorem, where possible rational roots are ±(factors of constant)/(factors of leading coefficient).