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Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 775 · 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 102 to make it divisible by 7? (a) 3 (b) 4 (c) 5 (d) 2
  2. 2.What is the smallest number you must add to 933 to make it divisible by 12? (a) 9 (b) 4 (c) 3 (d) 2
  3. 3.What is the smallest number you must add to 329 to make it divisible by 9? (a) 3 (b) 5 (c) 4 (d) 6
  4. 4.What is the remainder when 669 is divided by 12? (a) 9 (b) 3 (c) 10 (d) 8
  5. 5.What is the remainder when 188 is divided by 3? (a) 2 (b) 0 (c) 1
  6. 6.What is the remainder when 411 is divided by 8? (a) 2 (b) 4 (c) 3 (d) 5
  7. 7.What is the remainder when 105 is divided by 4? (a) 2 (b) 1 (c) 0 (d) 3
  8. 8.What is the units digit of 3^44? (a) 7 (b) 9 (c) 1 (d) 3
  9. 9.What is the units digit of 2^24? (a) 6 (b) 8 (c) 2 (d) 4
  10. 10.What is the remainder when 740 is divided by 12? (a) 8 (b) 7 (c) 4 (d) 9
  11. 11.What is the smallest number you must add to 46 to make it divisible by 4? (a) 1 (b) 2 (c) 3
  12. 12.Which of these numbers is divisible by 4? (a) 1245 (b) 1243 (c) 1244 (d) 1249
  13. 13.What is the units digit of 2^28? (a) 8 (b) 6 (c) 4 (d) 2
  14. 14.What is the smallest number you must add to 509 to make it divisible by 7? (a) 5 (b) 3 (c) 1 (d) 2
  15. 15.What is the units digit of 9^27? (a) 1 (b) 7 (c) 9 (d) 8
  16. 16.Which of these numbers is divisible by 8? (a) 6107 (b) 6118 (c) 6099 (d) 6104
  17. 17.What is the smallest number you must add to 281 to make it divisible by 6? (a) 1 (b) 2 (c) 5 (d) 3
  18. 18.Which of these numbers is divisible by 11? (a) 3051 (b) 3064 (c) 3058 (d) 3075
  19. 19.Which of these numbers is divisible by 8? (a) 2869 (b) 2851 (c) 2856 (d) 2857
  20. 20.What is the remainder when 573 is divided by 8? (a) 5 (b) 6 (c) 4 (d) 3
  21. 21.What is the remainder when 354 is divided by 5? (a) 3 (b) 4 (c) 1 (d) 2
  22. 22.What is the smallest number you must add to 743 to make it divisible by 10? (a) 3 (b) 7 (c) 8 (d) 6
  23. 23.What is the smallest number you must add to 683 to make it divisible by 10? (a) 8 (b) 3 (c) 6 (d) 7
  24. 24.Which of these numbers is divisible by 11? (a) 7469 (b) 7478 (c) 7468 (d) 7488
  25. 25.What is the remainder when 212 is divided by 7? (a) 5 (b) 1 (c) 2 (d) 3
  26. 26.What is the smallest number you must add to 128 to make it divisible by 5? (a) 2 (b) 3 (c) 4 (d) 1
  27. 27.Which of these numbers is divisible by 9? (a) 7994 (b) 7992 (c) 7991 (d) 8004
  28. 28.What is the smallest number you must add to 97 to make it divisible by 6? (a) 5 (b) 4 (c) 3 (d) 1
  29. 29.Which of these numbers is divisible by 6? (a) 1966 (b) 1975 (c) 1971 (d) 1968
  30. 30.What is the smallest number you must add to 406 to make it divisible by 6? (a) 1 (b) 3 (c) 4 (d) 2
  31. 31.What is the units digit of 2^59? (a) 8 (b) 4 (c) 2 (d) 6
  32. 32.What is the units digit of 2^27? (a) 4 (b) 2 (c) 6 (d) 8
  33. 33.Which of these numbers is divisible by 3? (a) 328 (b) 334 (c) 332 (d) 330
  34. 34.What is the smallest number you must add to 401 to make it divisible by 11? (a) 5 (b) 8 (c) 6 (d) 7
  35. 35.What is the units digit of 9^24? (a) 2 (b) 0 (c) 1 (d) 9
  36. 36.What is the remainder when 183 is divided by 5? (a) 1 (b) 2 (c) 3 (d) 4
  37. 37.Which of these numbers is divisible by 6? (a) 1748 (b) 1740 (c) 1743 (d) 1738
  38. 38.What is the remainder when 528 is divided by 7? (a) 3 (b) 4 (c) 5 (d) 2
  39. 39.What is the smallest number you must add to 107 to make it divisible by 4? (a) 3 (b) 2 (c) 1
  40. 40.What is the units digit of 9^23? (a) 1 (b) 7 (c) 9 (d) 8

Answer key

  1. 3
  2. 3
  3. 4
  4. 9
  5. 2
  6. 3
  7. 1
  8. 1
  9. 6
  10. 8
  11. 2
  12. 1244
  13. 6
  14. 2
  15. 9
  16. 6104
  17. 1
  18. 3058
  19. 2856
  20. 5
  21. 4
  22. 7
  23. 7
  24. 7469
  25. 2
  26. 2
  27. 7992
  28. 5
  29. 1968
  30. 2
  31. 8
  32. 8
  33. 330
  34. 6
  35. 1
  36. 3
  37. 1740
  38. 3
  39. 1
  40. 9

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