Finding factors of a number

A factor of a number is any whole number that divides it exactly, with no remainder. The job is to list every factor — without missing any.

Start with 1, work up

For any number, 1 and the number itself are always factors. Start there, then check 2, 3, 4, …

Factors of 24:

So the factors of 24 are: 1, 2, 3, 4, 6, 8, 12, 24 — eight factors, four pairs.

You can stop at the square root

Notice in the table that the pairs are symmetric: 1×24, 2×12, 3×8, 4×6. Once the two factors in the pair get close to each other, you're about to start repeating.

A neat shortcut: stop checking once you reach $\sqrt{n}$.

• For 24, $\sqrt{24}$ ≈ 4.9. So you only need to test 1, 2, 3, 4. Every factor you find before that has a "partner" larger than 4.9 — you don't need to look for it separately.
• For 36, $\sqrt{36}=6$ exactly. Test 1, 2, 3, 4, 5, 6. When 6 × 6 = 36 you've reached the middle of the pairs.

Square numbers have an odd number of factors

When a number is a perfect square (like 36), one of its factor pairs is a number times itself (6 × 6). That gives an odd total count of factors.

Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36 — nine factors.

Other numbers have an even count, because every factor has a different partner.

Quick divisibility tricks

For 24: 2 + 4 = 6, which is a multiple of 3, so 24 is divisible by 3. ✓

What's next

• Multiples and times tables
• Prime and composite numbers
• Back to the introduction

Try it out

• 🔢 List all factors of a number
• 🧐 Prime or composite? — practice spotting numbers with only two factors