Learning goal
Use this note to test divisibility, find all permitted missing digits and avoid rule-mixing. Divisibility means division leaves remainder 0.
The three rules
By 3: A number is divisible by 3 exactly when the sum of all its digits is divisible by 3.
By 9: A number is divisible by 9 exactly when its digit sum is divisible by 9. You may repeat digit addition for a large sum.
The connection: Divisible by 9 implies divisible by 3 because 9 = 3 × 3. The reverse is false: 12 is divisible by 3 but not by 9.
By 11: From the left, add digits in positions 1, 3, 5, …; separately add those in positions 2, 4, 6, …. Subtract the second sum from the first. The number is divisible by 11 exactly when this difference is 0 or another multiple of 11: 11, -11, 22, -22, ….
Zero works because 0 = 11 × 0. Negative multiples also work. Starting from the other end can reverse the difference’s sign but never changes divisibility. Alternate positions consistently; do not group digits by whether their values are odd or even.
A missing digit ranges from 0 to 9; a leading digit cannot be 0 in a stated multi-digit number. Letters within a number represent digits, not multiplication.
Four worked examples
Example 1 — Testing 3 and 9: 5724
Digit sum = 5 + 7 + 2 + 4 = 18. Therefore, 5724 is divisible by both 3 and 9. Checks: 5724 ÷ 3 = 1908; 5724 ÷ 9 = 636.
Example 2 — Zero difference: 38126
Difference = (3 + 1 + 6) - (8 + 2) = 0. Therefore, 38126 is divisible by 11. Check: 38126 ÷ 11 = 3466.
Example 3 — Negative difference: 2728
Difference = (2 + 2) - (7 + 8) = -11. This is a multiple of 11, so the test succeeds. Check: 2728 ÷ 11 = 248.
Example 4 — All missing-digit answers
Find every digit x making 63x9 divisible by 9. Its digit sum is 18 + x, ranging from 18 to 27. The multiples of 9 here are 18 and 27. Thus x = 0 or 9, giving 6309 and 6399. No other digit works.
Common mistakes
Do not test only the last digit or use an ordinary digit sum for 11. Keep zero digits in their positions when alternating.
Do not reject zero or negative differences. Divisibility by 3 does not prove divisibility by 9.
For missing digits, check every permitted multiple and exclude invalid leading zeros. Substitute answers back; when two conditions are stated, satisfy both.
Practice questions
Choose exactly one answer per question.
1. Which number is divisible by 3 but not by 9? A) 4518; B) 4521; C) 4523; D) 4525.
2. Which digit x makes 72x4 divisible by 9? A) 2; B) 3; C) 4; D) 5.
3. Which digit x makes 6x38 divisible by 11? A) 1; B) 2; C) 5; D) 9.
4. Which number is divisible by 11? A) 2817; B) 2819; C) 2827; D) 2829.
5. How many digits x make x42 a three-digit number divisible by 3? A) 2; B) 3; C) 4; D) 5.
6. For digits x and y, 5x4y is divisible by both 9 and 11. What is x + y? A) 0; B) 3; C) 9; D) 18.
Answers and explanations
1. B — 4521. Digit sums are 18, 12, 14 and 16, respectively. Only 12 is divisible by 3 but not by 9.
2. D — 5. The sum 13 + x ranges from 13 to 22. Its only multiple of 9 is 18, so x = 5.
3. A — 1. The difference is (6 + 3) - (x + 8) = 1 - x. Between -8 and 1, the only multiple of 11 is 0, giving x = 1.
4. C — 2827. Alternating differences are -12, -14, -11 and -13, respectively. Only -11 is a multiple of 11.
5. B — 3. The sum is x + 6. Valid digits are 3, 6 and 9. Zero would satisfy the sum test but would not produce a three-digit number.
6. C — 9. For 11, the difference 9 - (x + y) lies between -9 and 9. Only 0 qualifies, giving x + y = 9. The digit sum is then 18, satisfying the 9-test too.