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.
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.
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
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.
√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.
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 Questions
- Find all factors of 48.
- Find all factors of 100.
- How many factors does 64 have?
- Use divisibility rules to decide: is 4 a factor of 2,346?
- Find all factors of 1.