Divisibility Tests and Missing Digits

Learning outcome

Apply nine divisibility tests and find every valid missing digit without performing full division.

Rules and why they work

Divisibility means remainder 0. For 2, the last digit must be 0, 2, 4, 6 or 8; for 5, it must be 0 or 5; for 10, it must be 0. The remaining part is a multiple of 10.

For 4, test the number formed by the last two digits; for 8, the last three. The remaining part is a multiple of 100 or 1000, divisible by 4 or 8 respectively. For shorter numbers, test the whole number.

For 3 or 9, test the digit sum. Each place value 1, 10, 100, … leaves remainder 1 for either divisor, so the number and its digit sum differ by a multiple of that divisor. Divisibility by 9 implies divisibility by 3, not conversely.

For 6, pass both the 2-test and 3-test. An even multiple of 3 requires an even second factor, making it divisible by 6.

For 11, starting from the left, subtract the sum at positions 2, 4, … from the sum at positions 1, 3, …. Accept 0 or any positive or negative multiple of 11. Place values 1, 10, 100, 1000, … alternate one above and one below multiples of 11.

Worked examples

Example 1. Test 73512 for 6 and 8. It is even and its digit sum is 18, so it is divisible by 6. Its ending 512 = 8 × 64 proves divisibility by 8.

Example 2. In 4x32, x is one digit. For divisibility by 9, the sum 9 + x must be 9 or 18 because 0 ≤ x ≤ 9. Thus x = 0 or 9.

Example 3. For 1837, the alternating difference is (1 + 3) - (8 + 7) = -11. Thus 1837 is divisible by 11.

Common traps

Keep zeros in position. A multi-digit number cannot start with 0. Passing only the 2-test or only the 3-test does not prove divisibility by 6.

Practice questions

1. Which tests does 6420 pass among 2, 3, 4, 5, 6, 8, 9, 10, 11?
2. Find every digit x making 52x divisible by 4.
3. Find the digit x making 4x73 divisible by 11.

Answers and explanations

1. It passes 2, 3, 4, 5, 6, 10. Final digit 0 establishes divisibility by 2, 5, 10. Sum 12 establishes divisibility by 3; being even too, it passes 6. Ending 20 passes 4. But 420 fails 8, sum 12 fails 9, and alternating difference 4 fails 11.
2. The last two digits form 20 + x. Multiples of 4 from 20 to 29 are 20, 24, 28, so x = 0, 4, 8.
3. The difference 8 - x lies between -1 and 8. Only 0 qualifies, so x = 8.