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Learn Index · Number Sense and Numeration Grade 6–8

Divisibility rules

Quick tests that show whether a number divides evenly.

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Quick tests for division

A number is divisible by another if it divides with no remainder. Divisibility rules are shortcuts that let you test this without doing long division.

The rules

Divisible by Rule Example
2 Last digit is even (0, 2, 4, 6, 8) 348 ✓
3 Digit sum is divisible by 3 123 → 1 + 2 + 3 = 6 ✓
4 Last two digits form a number divisible by 4 1,316 → 16 ✓
5 Last digit is 0 or 5 2,875 ✓
6 Divisible by both 2 and 3 252 → even, digit sum 9 ✓
8 Last three digits form a number divisible by 8 7,120 → 120 ÷ 8 = 15 ✓
9 Digit sum is divisible by 9 4,725 → 18 ✓
10 Last digit is 0 930 ✓
11 Alternating sum of digits is divisible by 11 2,794 → 2 − 7 + 9 − 4 = 0 ✓

Why do these work?

  • 2, 5 and 10: Every 10 is divisible by 2, 5 and 10, so only the ones digit matters.
  • 4 and 8: Every 100 is divisible by 4, and every 1,000 is divisible by 8.
  • 3 and 9: Each place value (10, 100, 1,000) is 1 more than a multiple of 9 (9, 99, 999). So a number leaves the same remainder as its digit sum when divided by 3 or 9.

Testing for 7

There’s no easy rule for 7. One trick: double the last digit and subtract it from the rest. Repeat until the number is small.

  • 343 → 34 − (2 × 3) = 28. 28 = 4 × 7 ✓, so 343 is divisible by 7.

Using the rules

Divisibility rules help with:

  • Finding factors quickly.
  • Simplifying fractions: 135/225, both digit sums divisible by 9 → divide by 9 → 15/25 → 3/5.
  • Prime testing.
  • Mental maths and checking.

Worked examples

Example 1: Is 123 divisible by 3?

1 + 2 + 3 = 6, divisible by 3. Yes.

Example 2: Is 5,634 divisible by 6?

Even ✓; digit sum 18 ✓ (divisible by 3). Yes.

Example 3: Which of 2, 3, 4, 5, 9, 10 divide 3,240?

  • 2 ✓ (ends in 0)
  • 3 ✓ (digit sum 9)
  • 4 ✓ (40 ÷ 4 = 10)
  • 5 ✓
  • 9 ✓
  • 10 ✓

All of them!

Common mistakes

  • Using the last digit to test 3 or 9. For those, add all the digits.
  • Testing 6 with only one rule. It must pass both the 2 and 3 tests.
  • Using only the last digit for 4. Look at the last two digits.

Try it yourself

  1. Is 4,512 divisible by 4?
  2. Is 7,281 divisible by 9?
  3. Is 1,035 divisible by 6?
  4. List all of 2, 3, 5, 9, 10 that divide 450.

Answers: 1) yes (12) 2) yes (digit sum 18) 3) no (it’s odd) 4) all of them

Quick recap

  • Divisibility rules test division without dividing.
  • Use the last digit for 2, 5, 10; the digit sum for 3 and 9.
  • Combine rules for 6 (2 and 3).