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Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 996 · 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 185 to make it divisible by 7? (a) 3 (b) 5 (c) 4 (d) 6
  2. 2.Which of these numbers is divisible by 3? (a) 1951 (b) 1949 (c) 1945 (d) 1947
  3. 3.What is the remainder when 577 is divided by 11? (a) 5 (b) 7 (c) 6 (d) 4
  4. 4.What is the smallest number you must add to 597 to make it divisible by 8? (a) 4 (b) 5 (c) 2 (d) 3
  5. 5.What is the units digit of 2^13? (a) 2 (b) 8 (c) 4 (d) 6
  6. 6.What is the units digit of 9^23? (a) 7 (b) 8 (c) 9 (d) 1
  7. 7.Which of these numbers is divisible by 4? (a) 208 (b) 214 (c) 210 (d) 207
  8. 8.What is the smallest number you must add to 146 to make it divisible by 5? (a) 3 (b) 2 (c) 4 (d) 1
  9. 9.What is the smallest number you must add to 151 to make it divisible by 4? (a) 2 (b) 1 (c) 3
  10. 10.What is the smallest number you must add to 367 to make it divisible by 6? (a) 3 (b) 4 (c) 1 (d) 5
  11. 11.Which of these numbers is divisible by 4? (a) 473 (b) 481 (c) 479 (d) 476
  12. 12.What is the remainder when 316 is divided by 10? (a) 7 (b) 4 (c) 6 (d) 5
  13. 13.What is the remainder when 427 is divided by 6? (a) 1 (b) 2 (c) 5 (d) 0
  14. 14.Which of these numbers is divisible by 8? (a) 315 (b) 320 (c) 327 (d) 334
  15. 15.Which of these numbers is divisible by 3? (a) 1012 (b) 1009 (c) 1007 (d) 1008
  16. 16.What is the smallest number you must add to 915 to make it divisible by 12? (a) 8 (b) 10 (c) 9 (d) 3
  17. 17.Which of these numbers is divisible by 9? (a) 1053 (b) 1055 (c) 1048 (d) 1063
  18. 18.What is the units digit of 7^50? (a) 1 (b) 3 (c) 9 (d) 7
  19. 19.What is the units digit of 2^12? (a) 2 (b) 8 (c) 6 (d) 4
  20. 20.Which of these numbers is divisible by 4? (a) 2208 (b) 2214 (c) 2209 (d) 2205
  21. 21.What is the remainder when 693 is divided by 12? (a) 3 (b) 8 (c) 9 (d) 10
  22. 22.What is the smallest number you must add to 372 to make it divisible by 10? (a) 8 (b) 7 (c) 2 (d) 9
  23. 23.Which of these numbers is divisible by 4? (a) 559 (b) 561 (c) 555 (d) 556
  24. 24.What is the units digit of 7^42? (a) 9 (b) 7 (c) 1 (d) 3
  25. 25.What is the smallest number you must add to 361 to make it divisible by 11? (a) 9 (b) 1 (c) 2 (d) 3
  26. 26.Which of these numbers is divisible by 6? (a) 1430 (b) 1425 (c) 1435 (d) 1428
  27. 27.What is the smallest number you must add to 265 to make it divisible by 7? (a) 3 (b) 1 (c) 2 (d) 6
  28. 28.What is the smallest number you must add to 425 to make it divisible by 9? (a) 6 (b) 8 (c) 2 (d) 7
  29. 29.What is the smallest number you must add to 222 to make it divisible by 4? (a) 1 (b) 3 (c) 2
  30. 30.What is the smallest number you must add to 242 to make it divisible by 4? (a) 3 (b) 2 (c) 1
  31. 31.Which of these numbers is divisible by 11? (a) 4510 (b) 4523 (c) 4500 (d) 4518
  32. 32.Which of these numbers is divisible by 3? (a) 2649 (b) 2654 (c) 2647 (d) 2650
  33. 33.Which of these numbers is divisible by 6? (a) 450 (b) 458 (c) 449 (d) 452
  34. 34.What is the remainder when 232 is divided by 3? (a) 1 (b) 0 (c) 2
  35. 35.Which of these numbers is divisible by 6? (a) 4216 (b) 4202 (c) 4206 (d) 4207
  36. 36.Which of these numbers is divisible by 4? (a) 2967 (b) 2968 (c) 2970 (d) 2973
  37. 37.Which of these numbers is divisible by 4? (a) 3540 (b) 3546 (c) 3538 (d) 3542
  38. 38.Which of these numbers is divisible by 6? (a) 524 (b) 512 (c) 521 (d) 516
  39. 39.What is the remainder when 157 is divided by 4? (a) 2 (b) 3 (c) 0 (d) 1
  40. 40.Which of these numbers is divisible by 4? (a) 2193 (b) 2191 (c) 2197 (d) 2192

Answer key

  1. 4
  2. 1947
  3. 5
  4. 3
  5. 2
  6. 9
  7. 208
  8. 4
  9. 1
  10. 5
  11. 476
  12. 6
  13. 1
  14. 320
  15. 1008
  16. 9
  17. 1053
  18. 9
  19. 6
  20. 2208
  21. 9
  22. 8
  23. 556
  24. 9
  25. 2
  26. 1428
  27. 1
  28. 7
  29. 2
  30. 2
  31. 4510
  32. 2649
  33. 450
  34. 1
  35. 4206
  36. 2968
  37. 3540
  38. 516
  39. 1
  40. 2192

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