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Factors Calculator

Find every factor of any whole number — displayed as factor pairs, with a complete prime factorisation worked out step by step using the division ladder. See whether the number is prime, how many factors it has, and how it breaks down into primes.

Factors calculator

Works for any positive integer up to 1,000,000
1 is neither prime nor composite
There are:
— composite factors
— prime factors

Factor pairs

Prime factorisation — division ladder

Factors Calculator Explained

Core concepts, classroom applications, and plain-English explanations for all audiences.

Where factors appear in real life

You're tiling a bathroom with 24 tiles — how can you arrange them in a rectangle? The factor pairs of 24 give every option: 1×24, 2×12, 3×8, 4×6. Organising 30 children into equal groups? Find the factors of 30. Simplifying the fraction 18/24? You need to know that 6 divides both evenly.

Worked example

"Find all factors of 36."

Test 1: 36 ÷ 1 = 36 ✓ → pair: 1, 36

Test 2: 36 ÷ 2 = 18 ✓ → pair: 2, 18

Test 3: 36 ÷ 3 = 12 ✓ → pair: 3, 12

Test 4: 36 ÷ 4 = 9 ✓ → pair: 4, 9

Test 5: 36 ÷ 5 = 7.2 ✗

Test 6: 36 ÷ 6 = 6 ✓ → pair: 6, 6 (√36)

Result: 1, 2, 3, 4, 6, 9, 12, 18, 36

Stop testing at √36 = 6 — all factor pairs have already been found.

Common mistake to avoid

  • ✗ Missing the larger factor in each pair: if you find that 3 is a factor of 36, then 12 automatically is too (3 × 12 = 36). Always write down both numbers in a factor pair.
Good to know

Understanding factors

What is a factor?
A factor of a number divides into it exactly — with no remainder. Every whole number has at least two factors: 1 and itself. If a number has only those two factors, it is prime. The number 1 is special — it has exactly one factor (itself) and is neither prime nor composite.
Factors of 12: 1, 2, 3, 4, 6, 12
12 ÷ 1 = 12 ✓   12 ÷ 2 = 6 ✓
12 ÷ 3 = 4 ✓   12 ÷ 4 = 3 ✓
12 ÷ 5 = 2.4 ✗ (not exact)
Factor pairs
Every factor has a partner — the number you multiply it by to get back to the original. Finding all factors efficiently means stopping at √n: once you pass the square root, you've already seen the other half of every pair in reverse. That's why the calculator only tests up to √n and generates pairs from there.
Factors of 36 as pairs:
1 × 36   2 × 18   3 × 12
4 × 9   6 × 6 ← square root

√36 = 6, so we stop at 6
Prime factorisation
Every whole number greater than 1 can be written as a unique product of prime numbers — this is called the Fundamental Theorem of Arithmetic. The division ladder divides out the smallest prime factor at each step until only 1 remains. The primes used (with repetitions) are the prime factorisation.
60 ÷ 2 = 30
30 ÷ 2 = 15
15 ÷ 3 = 5
5 ÷ 5 = 1
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60 = 2² × 3 × 5
Interesting fact
Perfect squares have an odd number of factors
Factors come in pairs, so most numbers have an even count. But perfect squares (4, 9, 16, 25, 36…) have one factor that pairs with itself — the square root. That unpaired factor makes the total odd. This is a quick way to identify perfect squares without calculating: if the factor count is odd, it's a perfect square.
Factors of 36: 1,2,3,4,6,9,12,18,36
Count: 9 (odd) → perfect square ✓

Factors of 12: 1,2,3,4,6,12
Count: 6 (even) → not a perfect square