See every divisor of any number, its prime factorization step by step, the count and sum of its divisors, and whether it is prime, perfect, abundant or a perfect square.
A divisor of n is a whole number that divides n with no remainder. The divisors of 36 are 1, 2, 3, 4, 6, 9, 12, 18 and 36. The calculator finds them all by factoring the number into primes: 36 = 2² × 3². Adding 1 to each exponent and multiplying gives (2 + 1) × (2 + 1) = 9, the count of divisors. The sum is the product of the sums of the powers of each prime: (1 + 2 + 4) × (1 + 3 + 9) = 7 × 13 = 91. Leaving out 36 itself, the proper divisors add up to 55, which is more than 36, so 36 is an abundant number. If that sum equalled the number it would be perfect, like 6 (1 + 2 + 3) and 28 (1 + 2 + 4 + 7 + 14); if it were smaller, the number is deficient, as every prime is. The number 1 has one divisor and primes have exactly two. Perfect squares, such as 36 and 100, are the only numbers with an odd count of divisors, because the root pairs up with itself. The Divisibility tab applies the rules for 2, 3, 4, 5, 6, 8, 9, 10 and 11 and checks each against the real division. The Compare tab shows the GCD, the LCM and the shared divisors of two numbers: 36 and 48 have GCD 12 and LCM 144. The limit is 18 digits. For numbers that size the factoring uses the Miller-Rabin primality test and Pollard's rho algorithm, all with exact integers (BigInt), with no rounding. The number with the most divisors below 10^18 is 897,612,484,786,617,600, with 103,680. Zero has no finite list because every integer divides 0, and negative numbers share the divisors of their absolute value, also taken as negative.
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There are two ways. The direct one is to test the whole numbers from 1 up to the square root of the number: each time the division is exact you gain two divisors, the tested one and the quotient. For 36, testing 1, 2, 3, 4, 5 and 6 gives the pairs (1, 36), (2, 18), (3, 12), (4, 9) and (6, 6). Factoring is faster for large numbers: break the number into primes and combine the powers, which is what the Step by step tab shows.
36 has 9 divisors
The proper divisors add up to 55, which is 19 more than 36: an abundant number.
Divisors that are prime are highlighted.
Accepts whole numbers of up to 18 digits. Everything runs in your browser with exact arithmetic.
Related: GCD and LCM · Prime numbers · Simplify fractions