Divisibility rules and missing digits: practice sheet

Part 1: Last-digit rules: 2, 5, 10, 4 and 8

Remember: 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.

1. 1. Warm-up Is 3,458 divisible by 4? (yes or no)

2. 2. Warm-up Is 7,250 divisible by 10? (yes or no)

3. 3. Core The smallest number to add to 3,457 to make it divisible by 4

4. 4. Core Is 9,016 divisible by 8? (yes or no)

5. 5. Stretch 🔥 The largest digit d so that 23d4 divides by 8

Part 2: Digit sums: 3 and 9

Remember: A number divides by 3 when the sum of its digits divides by 3, and by 9 when the sum divides by 9.

1. 6. Warm-up Is 58,312 divisible by 3? (yes or no)

2. 7. Warm-up The remainder when 2,345 is divided by 9

3. 8. Core The smallest digit d so that 7d8 divides by 9

4. 9. Core The largest 3-digit number that divides by 9

5. 10. Stretch 🔥 How many digits d (0 to 9) make 5d1 divisible by 3?

Part 3: 11, and combining rules for 6 and 12

Remember: 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.

1. 11. Warm-up Is 8,349 divisible by 11? (yes or no)

2. 12. Warm-up Is 4,236 divisible by 6? (yes or no)

3. 13. Core Is 3,624 divisible by 12? (yes or no)

4. 14. Core The digit d so that 9d42 divides by 11

5. 15. Stretch 🔥 The smallest 4-digit number that divides by 11

Part 4: Missing-digit puzzles

Remember: 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.

1. 16. Warm-up The digit d so that 21d divides by 9

2. 17. Core The smallest digit d so that 6d5 divides by 15

3. 18. Core How many 2-digit numbers divide by 6?

4. 19. Stretch 🔥 The smallest number bigger than 1 that leaves remainder 1 when divided by 2, 3, 4, 5 and 6

Activity 1: Digit detective

Type a digit or a number for each one and press Check.

1. 20. Smallest digit d so that 3d5 divides by 3

2. 21. Digit d so that 52d4 divides by 9

3. 22. Does 123,456 divide by 8? (yes or no)

4. 23. Digit d so that 7d6 divides by 11

5. 24. Smallest 4-digit number that divides by 9

6. 25. Largest 3-digit number that divides by both 4 and 5

Activity 2: Tester investigation

Use the divisibility tester (or your rules) to find these numbers.
Steps: 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.

1. 26. Smallest 4-digit number divisible by 2, 3, 4, 5 and 6

2. 27. Smallest 4-digit number divisible by 9 and 11

3. 28. Smallest 4-digit number divisible by 8 and 9

4. 29. How many numbers from 1000 to 1100 divide by 36?

Activity 3: Secret number puzzles

Each puzzle has exactly one answer. Use the rules to cut down the choices.
Steps: 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.

1. 30. I am a 3-digit even number with hundreds digit 4. I divide by 5 and by 9. Who am I?

2. 31. The smallest 4-digit number that uses each of 1, 2, 3, 4 once and divides by 11

3. 32. The smallest number that divides by every number from 1 to 6

4. 33. The smallest number that divides by every number from 1 to 10

Homework

1. 34. Is 98,765 divisible by 5? By 3?

2. 35. Find the digit d so that 6d8 divides by 4 and by 3.

3. 36. Is 10,010 divisible by 11?

4. 37. What is the remainder when 5,555 is divided by 9?

Project: Divisibility detective card

Make a pocket card of the rules with your own examples, and use it to solve one big hunt.

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: ____________