Example 1
Is 3,716 divisible by 4?
- Only the last two digits matter: 16.
- 16 ÷ 4 = 4 exactly.
Answer: Yes
🏅 Olympiad maths · session 2 of 49 · 60 minutes
Divisibility rules and missing digits
Tell whether a big number divides exactly without doing the division. These rules make long questions short, and they are the key to the missing-digit puzzles that appear in every Olympiad.
A complete lesson · 60 minutes
Teach it, see it, practise it, all on this page.
Open this page with your learner and work down it, or press Teach this lesson to go through it one step at a time on a big screen. Every section has something to do.
- Warm-up4 min
- Last digits8 min
- Digit sums9 min
- 11, 6 and 127 min
- Missing digits7 min
- Activity 17 min
- Activity 27 min
- Activity 36 min
- Quiz4 min
- Set the project1 min
By the end you can
- Use the last-digit rules for 2, 4, 5, 8 and 10
- Use digit sums for 3 and 9, and explain why they work
- Use the rule for 11 and combine rules for 6 and 12
- Solve missing-digit puzzles
Before you start
- Session 1, or comfortable with multiplication tables
- Knowing what a remainder is
Have ready
- Paper and pencil
- A calculator only for checking at the end
Classes 4 to 7 · ages 9 to 13
Warm-up · think first
Without dividing, can 1,000,002 sweets be shared equally among 3 friends? How could you be sure?
Show the answer
Yes. 1 + 0 + 0 + 0 + 0 + 0 + 2 = 3, and when the digits add to a multiple of 3, the number is a multiple of 3. Today you will learn why that works.
Teacher tip: Let them try the long division first for a minute. Then show the digit trick and enjoy the surprise.
Part 1 of 4 · learn, see, practise
Last-digit rules: 2, 5, 10, 4 and 8
A number divides by 2 when its last digit is even, by 5 when it ends in 0 or 5, and by 10 when it ends in 0.
For 4, look at the last two digits only, because 100 is already a multiple of 4. For 8, look at the last three digits, because 1000 is a multiple of 8.
Example 2
Is 51,240 divisible by 8?
- Only the last three digits matter: 240.
- 240 ÷ 8 = 30 exactly.
Answer: Yes
Think first: Is 9,318 divisible by 4? By 2? Explain using the last digits.
Practise it
Start with the warm-up and go as far as you can. The stretch questions are meant to be hard.
Warm-up
Warm-up
Core
Core
Stretch 🔥
Answers (for the teacher)
- no
- yes
- 3
- yes
- 8
Part 2 of 4 · learn, see, practise
Digit sums: 3 and 9
A number divides by 3 when the sum of its digits divides by 3, and by 9 when the sum divides by 9.
Why? 10 is 9 + 1, 100 is 99 + 1, 1000 is 999 + 1. So 472 = 4 × 99 + 7 × 9 + (4 + 7 + 2). The parts with 9s and 99s are always multiples of 9 (and of 3), so only the digit sum decides.
Example 1
Is 7,482 divisible by 3? By 9?
- Digit sum: 7 + 4 + 8 + 2 = 21.
- 21 is a multiple of 3, so 7,482 divides by 3.
- 21 is not a multiple of 9, so it does not divide by 9.
Answer: By 3 yes, by 9 no
Think first: Find the remainder when 1,234 is divided by 9 using only the digit sum.
Practise it
Start with the warm-up and go as far as you can. The stretch questions are meant to be hard.
Warm-up
Warm-up
Core
Core
Stretch 🔥
Answers (for the teacher)
- no
- 5
- 3
- 999
- 4
Part 3 of 4 · learn, see, practise
11, and combining rules for 6 and 12
For 11, add and subtract the digits in turn from the left: + − + −. If the answer is 0, 11, −11 or any multiple of 11, the number divides by 11.
6 = 2 × 3, so a number divides by 6 when it passes the rules for 2 and for 3. 12 = 3 × 4, so check 3 and 4. (Use numbers with no common factor: checking 2 and 6 does not prove 12.)
Example 1
Is 9,163 divisible by 11?
- 9 − 1 + 6 − 3 = 11.
- 11 is a multiple of 11.
Answer: Yes (9,163 = 11 × 833)
Example 2
Is 2,316 divisible by 6?
- It ends in 6, which is even, so it divides by 2.
- 2 + 3 + 1 + 6 = 12, a multiple of 3, so it divides by 3.
- It passes both, so it divides by 6.
Answer: Yes
The divisibility tester
Change each digit. The table shows which numbers divide it and the rule that tells you.
- Make a 4-digit number that divides by both 9 and 4.
- Make an odd number that divides by 11.
- Make a 4-digit number that divides by 2, 3, 4, 5 and 6 at once.
Think first: Use the tester to find the smallest 4-digit number that divides by 11.
Practise it
Start with the warm-up and go as far as you can. The stretch questions are meant to be hard.
Warm-up
Warm-up
Core
Core
Stretch 🔥
Answers (for the teacher)
- yes
- yes
- yes
- 0
- 1001
Part 4 of 4 · learn, see, practise
Missing-digit puzzles
Olympiad questions hide a digit and ask what it must be. Write the rule with the unknown digit as a letter, then try the digits 0 to 9 that fit.
Always check your answer by dividing at the end.
Example 1
Find the digit d so that 4d72 divides by 9.
- Digit sum: 4 + d + 7 + 2 = 13 + d.
- The digit sum must be a multiple of 9. With d from 0 to 9, 13 + d is between 13 and 22, so it must be 18.
- d = 5. Check: 4,572 ÷ 9 = 508.
Answer: d = 5
Example 2
Find the digit d so that 35d divides by 6.
- For 2: d must be even.
- For 3: 3 + 5 + d = 8 + d must be 9, 12 or 15, so d is 1, 4 or 7.
- The only even choice is 4. Check: 354 ÷ 6 = 59.
Answer: d = 4
Think first: Find d so that 7d6 divides by 11.
Practise it
Start with the warm-up and go as far as you can. The stretch questions are meant to be hard.
Warm-up
Core
Core
Stretch 🔥
Answers (for the teacher)
- 6
- 1
- 15
- 61
Watch out
Common mistakes
Using the last digit for 4
Ends in 4 does not mean divides by 4: 14 does not. Use the last two digits.
Mixing up 3 and 9
A digit sum of 12 means divisible by 3 but not by 9.
Combining the wrong rules
Divisible by 2 and 4 does not prove divisible by 8. Combine rules only for numbers with no common factor.
Class activity 1 of 3
Digit detective
Type a digit or a number for each one and press Check.
Answers (for the teacher)
- 1
- 7
- yes
- 2
- 1008
- 980
Class activity 2 of 3
Tester investigation
Use the divisibility tester (or your rules) to find these numbers.
- A number divisible by 2, 3, 4, 5 and 6 divides by their LCM, 60.
- Divisible by 9 and 11 means divisible by 99; by 8 and 9 means by 72.
- Start from 1000 and find the first multiple.
- Count the multiples of 36 between 1000 and 1100.
Answers (for the teacher)
- 1020
- 1089
- 1008
- 3
Class activity 3 of 3
Secret number puzzles
Each puzzle has exactly one answer. Use the rules to cut down the choices.
- Write down what each clue tells you about the last digit or the digit sum.
- List the numbers that are left.
- Check your answer with every clue.
Answers (for the teacher)
- 450
- 1243
- 60
- 2520
Quiz · 8 questions
Check what you learned
Pick an answer to see if it is right and why. Your best score is saved on this device.
After class · project
Divisibility detective card
Make a pocket card of the rules with your own examples, and use it to solve one big hunt.
What to make: One card (or page) with all nine rules, examples, the explanation and the hunt answer with working.
Stretch: Find a rule for 7 on the internet or in a book and test it on three numbers.
How it is marked
| Look for | Done well when |
|---|---|
| Rules | All nine rules are correct |
| Examples | Every rule has a pass and a fail example |
| Why | The explanation for 9 uses 10 = 9 + 1 |
| Hunt | The smallest number is found with clear working |
Project template
Copy it onto paper or print the practice sheet.
Rule for 2: ______________ Passes: ____ Fails: ____ Rule for 3: ______________ Passes: ____ Fails: ____ … (one line for each of 4, 5, 6, 8, 9, 10, 11) Why the rule for 9 works: ________________________________ Smallest number divisible by 1 to 10: ________ How I found it: ____________
Homework
Practise at home
Try each on paper, then tap to see the answer and working. Or print the practice sheet.
Is 98,765 divisible by 5? By 3?
By 5 yes (ends in 5); by 3: 9+8+7+6+5 = 35, no.
Find the digit d so that 6d8 divides by 4 and by 3.
Last two digits d8 must divide by 4: d is even (08, 28, 48, 68, 88). Digit sum 14 + d must divide by 3: d = 1, 4 or 7. So d = 4 (648).
Is 10,010 divisible by 11?
1 − 0 + 0 − 1 + 0 = 0, so yes (10,010 = 11 × 910).
What is the remainder when 5,555 is divided by 9?
Digit sum 20, and 20 = 2 × 9 + 2, so the remainder is 2.
For parents and teachers
The 60-minute lesson plan
Use it as it is, or stretch any part over two sessions. Each row links to the part of the page you need.
| Time | Part | The teacher | The learner |
|---|---|---|---|
| 0–4 min | Warm-up | Ask the warm-up question and wait for a guess. | Guess, then reason. |
| 4–12 min | Last digits | Teach, show the examples step by step, then the practice. | Practise up to the stretch question. |
| 12–21 min | Digit sums | Teach, use the interactive part, then the practice. | Practise; try the stretch. |
| 21–28 min | 11, 6 and 12 | Teach and show the examples. | Practise. |
| 28–35 min | Missing digits | Teach and show the examples. | Practise; the stretch can be homework. |
| 35–42 min | Activity 1 | Let the learner work alone; hint only after a wrong check. | Solve and check. |
| 42–49 min | Activity 2 | Follow the steps together for the first item. | Finish the investigation. |
| 49–55 min | Activity 3 | Set as homework if time is short. | Solve the puzzles. |
| 55–59 min | Quiz | Run the quiz and review mistakes. | Answer and read why. |
| 59–60 min | Set the project | Show the template and the due date. | Start the project. |
Questions to ask
- Why do we only need the last two digits for 4?
- Why does the digit sum work for 9 but not for 7?
- Why is checking 2 and 4 not enough to prove 8?
If it feels hard
Stay with 2, 5, 10, 3 and 9 and use the tester to check every guess.
If it feels easy
Find all digits a and b so that the number 7a5b divides by 99 (it must divide by 9 and by 11).
FAQ
Divisibility rules and missing digits: questions
What are the divisibility rules for Olympiad maths?
2, 5 and 10 use the last digit; 4 uses the last two digits and 8 the last three; 3 and 9 use the digit sum; 11 uses the alternating sum; 6 and 12 combine two rules.
Why does the divisibility rule for 9 work?
Every place value is one more than a multiple of 9 (10 = 9 + 1, 100 = 99 + 1), so a number and its digit sum leave the same remainder when divided by 9.
How do you solve missing-digit divisibility questions?
Write the rule with the missing digit as a letter, list which digits 0 to 9 make it work, and check the answer by dividing.
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