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➗ Divisibility Rules 2 to 13 with Examples
A divisibility rule tells you whether one number divides another exactly — without doing the division. They are essential for simplifying fractions, finding factors, HCF and LCM, and they appear in almost every Maths Olympiad paper.
The rules
| Divisor | Rule | Example |
|---|---|---|
| 2 | Last digit is even (0, 2, 4, 6, 8) | 3,584 ✔ |
| 3 | Sum of digits divisible by 3 | 4,521 → 4+5+2+1 = 12 ✔ |
| 4 | Last two digits divisible by 4 | 7,316 → 16 ✔ |
| 5 | Last digit 0 or 5 | 8,935 ✔ |
| 6 | Divisible by both 2 and 3 | 1,254: even, digit sum 12 ✔ |
| 7 | Double the last digit and subtract it from the rest; repeat | 581 → 58 − 2 = 56 = 7 × 8 ✔ |
| 8 | Last three digits divisible by 8 | 9,128 → 128 = 8 × 16 ✔ |
| 9 | Sum of digits divisible by 9 | 6,543 → 18 ✔ |
| 10 | Last digit 0 | 4,570 ✔ |
| 11 | Alternating sum of digits is 0 or divisible by 11 | 9,163 → 9 − 1 + 6 − 3 = 11 ✔ |
| 12 | Divisible by both 3 and 4 | 5,724: digit sum 18, last two 24 ✔ |
| 13 | Add 4 × the last digit to the rest; repeat | 377 → 37 + 28 = 65 = 13 × 5 ✔ |
Why the rules for 3 and 9 work
Every power of 10 is one more than a multiple of 9 (10 = 9 + 1, 100 = 99 + 1 …). So a number like 4,521 is 4 × 1000 + 5 × 100 + 2 × 10 + 1, which leaves the same remainder on division by 9 (or 3) as 4 + 5 + 2 + 1. That is why the digit sum decides it.
Combining rules
For a divisor made of co-prime parts, test each part: divisible by 15 means divisible by 3 and 5; by 18 means by 2 and 9; by 24 means by 3 and 8. (Do not use 4 and 6 for 24 — they share a factor of 2, so 12 would pass both tests.)
Olympiad-style practice
- Find the digit k so that 5k4 is divisible by 9. → 5 + k + 4 must be 9 or 18, so k = 0 or 9.
- Is 2,728 divisible by 11? → 2 − 7 + 2 − 8 = −11 ✔ Yes.
- What is the smallest number to add to 1,234 to make it divisible by 3? → digit sum 10, next multiple of 3 is 12, so add 2.
Practise factors and multiples with instant feedback in BeadBop's School Maths (Classes 4–6) and the Olympiad Prep topic tests.
Quick answers
What is the divisibility rule of 7?
Double the last digit and subtract it from the number formed by the remaining digits. If the result is divisible by 7 (including 0), so is the original. Example: 581 → 58 − 2 = 56, which is 7 × 8.
What is the divisibility rule of 11?
Take the alternating sum of the digits from the left (first − second + third − …). If it is 0 or a multiple of 11, the number is divisible by 11.
How do you check divisibility by 4?
Look only at the last two digits. If they form a number divisible by 4, the whole number is divisible by 4.