Divisor Calculator — Find All Divisors of a Number
Enter any positive integer to instantly list all of its divisors, count them, compute the sum of divisors σ(N), and classify the number as perfect, abundant or deficient based on its proper divisors.
Divisors of 12: 1, 2, 3, 4, 6, 12
How does this calculator work?
A divisor of N is any integer d where N mod d = 0. The calculator checks all values from 1 to √N, collects divisor pairs, and sums them to give σ(N). Proper divisors (excluding N) sum to s(N): if s(N) = N the number is perfect, s(N) > N is abundant, and s(N) < N is deficient. Works for any positive integer up to 100 million.
Formula
How this is calculated
A divisor (or factor) of a positive integer N is any integer d ≥ 1 such that N mod d = 0. The calculator finds all divisors efficiently by checking every integer from 1 up to √N: if i divides N, then both i and N/i are divisors, so at most √N checks are needed. The divisors are then sorted in ascending order.
The sigma function σ(N) is the sum of all positive divisors, including 1 and N itself. The proper divisors of N are all divisors except N; their sum is s(N) = σ(N) − N. Number theorists classify integers by comparing s(N) to N: if s(N) = N the number is *perfect* (the smallest example is 6: 1+2+3 = 6); if s(N) > N it is *abundant* (e.g. 12: 1+2+3+4+6 = 16 > 12); if s(N) < N it is *deficient* (e.g. any prime, where s(p) = 1 < p).
This calculator handles positive integers up to 100,000,000 (10^8). At that limit the inner loop runs at most 10,000 iterations, which is fast in any modern browser. Negative inputs and zero are not valid for divisor analysis in the standard integer sense.
Frequently asked questions
Divisor and factor are synonyms in this context: both describe a positive integer d that divides N with no remainder. Some texts reserve 'factor' for the factors in a prime factorization, but in elementary number theory the two terms are used interchangeably.
A perfect number equals the sum of its proper divisors (all divisors except itself). The first four perfect numbers are 6, 28, 496, and 8128. Only 51 perfect numbers are currently known; all known examples are even, and it is an open question whether any odd perfect numbers exist.
If N = p₁^a₁ × p₂^a₂ × … (prime factorization), then the number of divisors is (a₁+1)(a₂+1)…. For example 12 = 2² × 3¹ has (2+1)(1+1) = 6 divisors. This formula gives the count directly without finding each divisor.
Also known as
TG we-Calculate Editorial Team. (2026). Divisor Calculator — Find All Divisors of a Number [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/divisor-calculator
TG we-Calculate Editorial Team. "Divisor Calculator — Find All Divisors of a Number." TG we-Calculate. 2026. https://we-calculate.com/calculator/divisor-calculator.
TG we-Calculate Editorial Team, "Divisor Calculator — Find All Divisors of a Number," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/divisor-calculator
@misc{wecalculate_divisor_calculator, title = {Divisor Calculator — Find All Divisors of a Number}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/divisor-calculator}}, year = {2026}, note = {TG we-Calculate} }
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