Divisors Calculator – Find Every Factor of a Number
A divisor (also called a factor) of an integer n is any positive whole number that divides n with no remainder. This calculator lists every divisor of a number instantly, groups them into factor pairs, sums the proper divisors, and classifies the number as perfect, abundant, or deficient. It also finds the common divisors shared between two numbers, making it useful for students, teachers, and anyone exploring number theory.
All Divisors vs. Proper Divisors
The full divisor list of a number includes the number itself. For example, the divisors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, 36 — nine divisors in total. The proper divisors exclude the number itself, leaving 1, 2, 3, 4, 6, 9, 12, 18. This distinction matters most when computing the sum used for the perfect/abundant/deficient classification below.
How Factor Pairs Work
Every divisor i of n has a matching cofactor n / i such that i × (n / i) = n. Grouping divisors this way produces factor pairs like (1, 36), (2, 18), (3, 12), (4, 9), (6, 6) for 36. Notice that when a number is a perfect square, the middle pair repeats the same value — 6 × 6 — since the square root divides evenly into itself.
Perfect, Abundant, and Deficient Numbers
Comparing a number to the sum of its proper divisors reveals a classic number-theory classification. If the sum equals the number, it is a perfect number — 28 is perfect because 1 + 2 + 4 + 7 + 14 = 28. If the sum exceeds the number, it is abundant — 12 is abundant because its proper divisors add up to 16. If the sum falls short, the number is deficient — every prime number is deficient since its only proper divisor is 1.
Finding Common Divisors of Two Numbers
The common divisors shared by two integers are exactly the divisors of their Greatest Common Divisor (GCD). For 24 and 36, the GCD is 12, so the common divisors are the divisors of 12: 1, 2, 3, 4, 6, 12. This calculator computes the GCD with the Euclidean algorithm first, then lists every divisor of that result.
The Trial Division Algorithm
Finding divisors by checking every integer from 1 to n would be slow for large numbers. Instead, this tool only tests integers up to √n. Whenever a value i divides n evenly, both i and its cofactor n / i are recorded as divisors — cutting the work from O(n) to roughly O(√n) steps. The prime factorization shown alongside the results uses the same principle to break the number into its prime building blocks.
720 or 831,600 have far more divisors than nearby values because they pack in many small prime factors raised to higher powers. A prime number, by contrast, has exactly two divisors: 1 and itself.Practical Uses
Beyond classroom number theory, divisor enumeration comes up when simplifying fractions, finding common denominators, scheduling repeating events, arranging items into equal rows and columns, and verifying divisor-related logic in programming assignments. The factor pairs are especially handy for factoring quadratic expressions and mental arithmetic.