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Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 751 · 40 questions
TalentJR

How to solve it (Patterns repeat): Divisibility: 3 and 9 — digit sum; 4 — last two digits; 8 — last three digits; 6 — even and divisible by 3; 11 — alternate digit sums differ by 0 or a multiple of 11. Remainder: find the biggest multiple not more than the number and take it away; to make a number divisible, add divisor − remainder. Units digit of a power: write the cycle (7, 9, 3, 1 for 7), divide the power by the cycle length and use the remainder.

NameDateTime takenScore ___ / 40
  1. 1.What is the smallest number you must add to 117 to make it divisible by 4? (a) 1 (b) 3 (c) 2
  2. 2.What is the smallest number you must add to 59 to make it divisible by 3? (a) 2 (b) 1
  3. 3.What is the smallest number you must add to 578 to make it divisible by 9? (a) 2 (b) 7 (c) 8 (d) 6
  4. 4.What is the remainder when 420 is divided by 9? (a) 5 (b) 7 (c) 3 (d) 6
  5. 5.Which of these numbers is divisible by 6? (a) 2496 (b) 2500 (c) 2505 (d) 2494
  6. 6.Which of these numbers is divisible by 8? (a) 5019 (b) 5013 (c) 5008 (d) 5002
  7. 7.What is the remainder when 108 is divided by 8? (a) 6 (b) 4 (c) 3 (d) 5
  8. 8.What is the remainder when 38 is divided by 3? (a) 0 (b) 2 (c) 1
  9. 9.What is the smallest number you must add to 463 to make it divisible by 7? (a) 6 (b) 5 (c) 4 (d) 1
  10. 10.What is the smallest number you must add to 245 to make it divisible by 6? (a) 5 (b) 3 (c) 2 (d) 1
  11. 11.What is the units digit of 4^37? (a) 6 (b) 5 (c) 4 (d) 3
  12. 12.What is the smallest number you must add to 227 to make it divisible by 9? (a) 8 (b) 2 (c) 6 (d) 7
  13. 13.Which of these numbers is divisible by 4? (a) 3442 (b) 3445 (c) 3440 (d) 3439
  14. 14.Which of these numbers is divisible by 11? (a) 3822 (b) 3828 (c) 3829 (d) 3849
  15. 15.What is the smallest number you must add to 359 to make it divisible by 5? (a) 1 (b) 2 (c) 3 (d) 4
  16. 16.Which of these numbers is divisible by 6? (a) 4914 (b) 4910 (c) 4923 (d) 4916
  17. 17.What is the units digit of 8^50? (a) 6 (b) 2 (c) 4 (d) 8
  18. 18.What is the smallest number you must add to 389 to make it divisible by 12? (a) 7 (b) 6 (c) 8 (d) 5
  19. 19.What is the smallest number you must add to 190 to make it divisible by 9? (a) 6 (b) 1 (c) 8 (d) 7
  20. 20.What is the remainder when 793 is divided by 10? (a) 2 (b) 7 (c) 3 (d) 4
  21. 21.What is the units digit of 3^11? (a) 3 (b) 1 (c) 9 (d) 7
  22. 22.What is the smallest number you must add to 242 to make it divisible by 4? (a) 1 (b) 2 (c) 3
  23. 23.Which of these numbers is divisible by 11? (a) 2193 (b) 2189 (c) 2183 (d) 2203
  24. 24.What is the remainder when 578 is divided by 12? (a) 1 (b) 10 (c) 2 (d) 3
  25. 25.What is the smallest number you must add to 390 to make it divisible by 12? (a) 8 (b) 5 (c) 7 (d) 6
  26. 26.What is the units digit of 3^48? (a) 9 (b) 7 (c) 1 (d) 3
  27. 27.What is the smallest number you must add to 229 to make it divisible by 4? (a) 2 (b) 3 (c) 1
  28. 28.What is the units digit of 9^45? (a) 8 (b) 1 (c) 9 (d) 7
  29. 29.What is the units digit of 4^42? (a) 4 (b) 7 (c) 5 (d) 6
  30. 30.Which of these numbers is divisible by 4? (a) 978 (b) 972 (c) 975 (d) 971
  31. 31.What is the smallest number you must add to 261 to make it divisible by 5? (a) 1 (b) 4 (c) 2 (d) 3
  32. 32.What is the units digit of 3^50? (a) 9 (b) 3 (c) 7 (d) 1
  33. 33.What is the remainder when 566 is divided by 8? (a) 7 (b) 2 (c) 6 (d) 5
  34. 34.Which of these numbers is divisible by 8? (a) 690 (b) 703 (c) 684 (d) 688
  35. 35.Which of these numbers is divisible by 9? (a) 4135 (b) 4140 (c) 4157 (d) 4142
  36. 36.What is the units digit of 2^16? (a) 4 (b) 8 (c) 6 (d) 2
  37. 37.What is the smallest number you must add to 369 to make it divisible by 7? (a) 2 (b) 3 (c) 1 (d) 5
  38. 38.What is the remainder when 92 is divided by 3? (a) 2 (b) 0 (c) 1
  39. 39.What is the remainder when 718 is divided by 9? (a) 2 (b) 7 (c) 6 (d) 8
  40. 40.Which of these numbers is divisible by 6? (a) 2689 (b) 2677 (c) 2687 (d) 2682

Answer key

  1. 3
  2. 1
  3. 7
  4. 6
  5. 2496
  6. 5008
  7. 4
  8. 2
  9. 6
  10. 1
  11. 4
  12. 7
  13. 3440
  14. 3828
  15. 1
  16. 4914
  17. 4
  18. 7
  19. 8
  20. 3
  21. 7
  22. 2
  23. 2189
  24. 2
  25. 6
  26. 1
  27. 3
  28. 9
  29. 6
  30. 972
  31. 4
  32. 9
  33. 6
  34. 688
  35. 4140
  36. 6
  37. 2
  38. 2
  39. 7
  40. 2682

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