Divisors Calculator

How to use

1. Enter an integer (1 or greater) whose divisors you want.
2. Press “Calculate”.
3. See the list of divisors, the count, the sum, pairs, prime factorization, and the count formula.

Examples

• Divisors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36 (9 in all, sum 91)
• 97 is prime, so its divisors are just 1 and 97 (2 in all)
• 6 is a perfect number (1, 2, 3, 6; the proper divisors 1+2+3 = 6)

When to use it (grade level)

Divisors (factors) are taught from around 5th grade and lead on to prime factorization, greatest common divisors, and least common multiples in middle school. This tool suits checking your factor-listing practice, reviewing worksheets, and confirming your work before a test.

A divisor of a number is an integer that divides it evenly. For example, the divisors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, and 36 — nine numbers that all divide 36 with no remainder. Listing them in order from smallest gives the full list of divisors.

How to list them yourself

The basic method is to try dividing by 1, 2, 3, and so on, but that gets slow for large numbers. A faster idea is to think in pairs that multiply to the original number. Divisors always come in pairs. For 36 the pairs are 1×36, 2×18, 3×12, 4×9, and 6×6, which unfold into the nine divisors 1, 2, 3, 4, 6, 9, 12, 18, 36.

You only need to check the smaller member of each pair up to the square root of the number. The square root of 36 is 6, so look among 1–6 for numbers that divide 36 (1, 2, 3, 4, 6) and write each partner (36, 18, 12, 9, 6). A pair of equal numbers like 6×6 counts as just one divisor.

Counting divisors without listing them

You can find how many divisors a number has without counting them one by one — just use its prime factorization. First break the number into primes: 36 = 2^2 × 3^2 (two 2s and two 3s).

The trick is to think of each prime as something you may use 0, 1, 2, … times. You can use the 2 zero, one, or two times (3 choices), and the 3 zero, one, or two times (3 choices). That gives 3×3 = 9 combinations, which is the number of divisors. As a formula, it is the product of each exponent plus one: (2+1)×(2+1) = 9.

For example, 100 = 2^2 × 5^2 gives (2+1)×(2+1) = 9, and 24 = 2^3 × 3 gives (3+1)×(1+1) = 8. Even for large numbers, once you can factorize you can count the divisors without writing them all out. The sum has a formula too, but mastering the count formula first is the best starting point.

Common mistakes

Three slip-ups are common. First, forgetting to include 1 and the number itself — every number has 1 and itself as divisors. Second, double-counting a pair for a perfect square: in 36 the pair 6×6 contributes just one divisor, 6.

Third, in the count formula, forgetting the “+1” and multiplying the exponents directly. Writing 36 = 2^2 × 3^2 as 2×2 = 4 divisors is wrong; the correct answer is (2+1)×(2+1) = 9. Always add one to each exponent before multiplying.

Related topics and tools

The count formula is built on prime factorization, so pairing this with the prime factorization calculator deepens your understanding. And since the largest shared divisor of two numbers is their greatest common divisor and the smallest shared multiple is their least common multiple, it also links to the GCD and LCM calculator. Try writing out the pairs yourself first, then check your answer here.

FAQ

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