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Factors Calculator — every divisor & factor pair

List every factor (divisor) of a whole number, see all factor pairs in a clean diagram, the total count and sum, and whether the number is prime — with an AI explanation of the pattern.

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Factors Calculator

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Quick answer

What is a factor?

A factor (or divisor) of a number is any whole number that divides it with no remainder. For example, the factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18 and 36. Factors always come in pairs that multiply to the original number (e.g. 4 × 9 = 36).

Why this tool

See the structure, not just a list

Factor pairs reveal symmetry around the square root — a pattern this tool draws for you.

Factor pairs visualized

Every pair that multiplies to your number is drawn, meeting at the square root.

Count & sum instantly

Get how many factors there are and their total — useful for divisor-sum problems.

Prime & square checks

Spot primes (only two factors) and perfect squares (an odd number of factors).

How it works

Find factors in three steps

Enter a number

Type any whole number, e.g. 36.

We pair to the square root

Each small divisor is matched with its partner number/divisor.

Read the full breakdown

Get every factor, the pairs diagram, count, sum and AI insight.

Worked examples

Factor examples

Tap any row to load it into the calculator.

Number	Factors	Count
36	1,2,3,4,6,9,12,18,36	9 (odd → square)
48	1,2,3,4,6,8,12,16,24,48	10
17	1, 17	2 → prime
100	1,2,4,5,10,20,25,50,100	9 (odd → square)
60	1,2,3,4,5,6,10,12,15,20,30,60	12
144	1…144	15 (odd → square)
24	1,2,3,4,6,8,12,24	8
12	1,2,3,4,6,12	6
18	1,2,3,6,9,18	6
72	1,2,3,4,6,8,9,12,18,24,36,72	12
64	1,2,4,8,16,32,64	7 (odd → square)

How to find all the factors of a number

The efficient way is to test every integer from 1 up to the square root of the number. Whenever one divides evenly, it forms a factor pair with its partner (the number ÷ that divisor). This finds every factor in roughly √n steps instead of n.

Worked example: 36

1×36, 2×18, 3×12, 4×9, 6×6 — so the factors are 1, 2, 3, 4, 6, 9, 12, 18, 36. Because 6×6 repeats, 36 has an odd number of factors — a signature of every perfect square.

Reading the patterns

Primes have exactly two factors: 1 and themselves.
Perfect squares have an odd number of factors.
The sum of factors appears in perfect-number and abundant-number problems.

The number of factors equals the product of (each prime exponent + 1) — link this with the prime factorization calculator.

Where factors are used

Arranging objects into equal rows or groups.
Finding common denominators and simplifying fractions.
Scheduling, tiling and packing problems.

FAQ

Frequently asked questions

What are the factors of a number?

Factors are the whole numbers that divide it exactly, with no remainder. For example, the factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18 and 36.

What is a factor pair?

A factor pair is two factors that multiply to give the number, such as 4 × 9 = 36. Listing pairs is the quickest way to find every factor.

Why do perfect squares have an odd number of factors?

Because one factor pair is a number multiplied by itself (e.g. 6 × 6 = 36), so it is only counted once — making the total count odd.

How is a factor different from a multiple?

A factor divides into the number; a multiple is the number multiplied by an integer. 4 is a factor of 36, while 72 is a multiple of 36.

Does the tool show the sum of factors?

Yes — it reports both the count and the sum of all factors, which is handy for perfect-number and divisor-sum questions.

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