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:
Try dividing 24 by
Does it work?
Pair
1
yes
1 × 24
2
yes (24 ÷ 2 = 12)
2 × 12
3
yes (24 ÷ 3 = 8)
3 × 8
4
yes (24 ÷ 4 = 6)
4 × 6
5
no (24 ÷ 5 leaves a remainder)
—
6
yes (24 ÷ 6 = 4) — already found!
—
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 n.
For 24, 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, 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
To test divisibility by
Trick
2
the number is even (ends in 0, 2, 4, 6, 8)
3
the sum of the digits is a multiple of 3
5
the number ends in 0 or 5
9
the sum of the digits is a multiple of 9
10
the number ends in 0
For 24: 2 + 4 = 6, which is a multiple of 3, so 24 is divisible by 3. ✓
