Factor Calculator
Find all factors, factor pairs, prime factorization, and the total number of factors for any positive integer.
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How to use this calculator
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Enter any positive integer up to 10,000,000.
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All factors are listed in ascending order, along with factor pairs.
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The prime factorization is displayed using exponent notation (e.g., 2^3 × 3^2).
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The calculator also tells you how many factors the number has and whether it is prime.
Frequently asked questions
What is the difference between a factor and a multiple?
A factor of n divides n evenly (n ÷ factor has no remainder). A multiple of n is n multiplied by an integer (n, 2n, 3n, …). Factors are finite in count; multiples are infinite. For example, factors of 12: {1, 2, 3, 4, 6, 12}; multiples of 12: {12, 24, 36, 48, …}.
How does prime factorization work?
Any integer greater than 1 can be written uniquely as a product of prime numbers (Fundamental Theorem of Arithmetic). To find it, divide repeatedly by the smallest prime (2, then 3, 5, 7, 11, …) until the quotient is 1. For 360: 360 = 2^3 × 3^2 × 5. Every factor of 360 is formed by choosing exponents for 2 (0–3), 3 (0–2), and 5 (0–1) — giving (3+1)(2+1)(1+1) = 24 factors total.
How do you find the number of factors from the prime factorization?
If n = p₁^a₁ × p₂^a₂ × … × pₖ^aₖ, then the number of factors is (a₁+1)(a₂+1)…(aₖ+1). For 360 = 2^3 × 3^2 × 5^1: (3+1)(2+1)(1+1) = 4 × 3 × 2 = 24 factors. This formula counts all combinations of prime factor powers from 0 up to each exponent.
What is a perfect number?
A perfect number equals the sum of its proper factors (all factors except itself). The smallest is 6 (factors 1+2+3 = 6), then 28 (1+2+4+7+14 = 28), then 496. All known perfect numbers are even and correspond to Mersenne primes — an open problem in mathematics is whether any odd perfect numbers exist.
Factor calculator — all factors, pairs, and prime factorization
Finding factors efficiently
The key insight is that factors come in pairs: if i is a factor of n, so is n/i. Therefore you only need to check divisors up to √n, which is much faster than checking all numbers up to n. For n = 1,000,000, you only need to check up to 1,000 (1,000 iterations instead of 1,000,000). This is the algorithm used here.
Prime factorization and the Fundamental Theorem of Arithmetic
Every integer greater than 1 factors uniquely into primes — this is the Fundamental Theorem of Arithmetic. The prime factorization of 360 = 2^3 × 3^2 × 5 encodes everything about its divisibility: 360 is divisible by 2 but not 4^2=16, by 9 but not 27, by 5 but not 25. GCD and LCM of two numbers can be read directly from their prime factorizations: GCD uses the minimum exponent for each prime; LCM uses the maximum.
Applications in number theory and cryptography
Factoring large numbers is computationally hard — this hardness is the foundation of RSA encryption. A 2048-bit RSA key uses a number n = p × q (product of two large primes, each ~617 digits). Factoring n to recover p and q would take longer than the age of the universe with current computers. Ironically, the same simple factoring algorithm used in this calculator, applied to numbers with hundreds of digits, is at the heart of modern internet security.
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