Finding Factors of a Number – Step-by-Step Methods

Finding all the factors of a number is a skill that becomes second nature with practice. There are two reliable methods: the division method and the square root shortcut.

Finding factors efficiently matters more than it might seem. The square root shortcut below is the classical “trial division” method — genuinely the oldest algorithm known for testing whether a number is prime, used by mathematicians for well over two thousand years and still the first line of attack computers use today before switching to more advanced techniques for very large numbers. Cutting the search from “every number up to N” down to “every number up to √N” is a huge saving: for a six-digit number like 123,456, it is the difference between testing over 123,000 possible divisors and testing only about 351.

Method 1 – Systematic Division

Divide the number by every whole number from 1 upward. Stop when you reach the number itself. Record every result that leaves remainder 0.

Find all factors of 28

28 ÷ 1 = 28 ✓ 28 ÷ 2 = 14 ✓ 28 ÷ 3 = 9.33 ✗ 28 ÷ 4 = 7 ✓ 28 ÷ 5 = 5.6 ✗ 28 ÷ 6 = 4.67 ✗ 28 ÷ 7 = 4 ✓ 28 ÷ 28 = 1 ✓

Factors: 1, 2, 4, 7, 14, 28

Watch the same idea run on 36: every whole number from 1 to 36 gets tested for divisibility, one batch at a time, colouring in green wherever it divides evenly.

Divides evenly — a factor

Leaves a remainder — not a factor

Final factor list

Test every whole number from 1 to 36

Click Step 1 or Auto Play to begin!

Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36
Nine numbers divided in evenly out of the 36 tested — those are all the factors.

Method 2 – Square Root Shortcut

You only need to test divisors up to the square root of the number. If a number divides in, both it and its partner are factors.

Find all factors of 36

√36 = 6. Test 1 to 6 only.

36 ÷ 1 = 36 ✓ (pair: 1, 36) 36 ÷ 2 = 18 ✓ (pair: 2, 18) 36 ÷ 3 = 12 ✓ (pair: 3, 12) 36 ÷ 4 = 9 ✓ (pair: 4, 9) 36 ÷ 5 = 7.2 ✗ 36 ÷ 6 = 6 ✓ (pair: 6, 6 — same number)

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

Method 3 – Factor Rainbow

Write the number at the top. Draw arcs connecting factor pairs. The rainbow fills inward until all pairs are found.

Factors of 12: 1—12, 2—6, 3—4

Divisibility Rules – Quick Reference

Divisor	Rule	Example
2	Last digit is even	48 ✓
3	Sum of digits divisible by 3	123: 1+2+3=6 ✓
4	Last two digits divisible by 4	316: 16÷4=4 ✓
5	Last digit 0 or 5	75 ✓
6	Divisible by both 2 and 3	48 ✓
9	Sum of digits divisible by 9	729: 7+2+9=18 ✓
10	Last digit is 0	130 ✓

Key Takeaways

Test divisors from 1 up to the square root — pairs fill in the rest.
Use divisibility rules to test quickly without long division.
Always list factors in ascending order.
A perfect square has an odd number of factors (the square root has no pair).

Practice: Find Factors

Factor-Finding Question