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Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 426 · 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.Which of these numbers is divisible by 6? (a) 513 (b) 504 (c) 505 (d) 500
  2. 2.What is the smallest number you must add to 133 to make it divisible by 10? (a) 3 (b) 6 (c) 7 (d) 8
  3. 3.What is the remainder when 410 is divided by 7? (a) 5 (b) 6 (c) 3 (d) 4
  4. 4.What is the remainder when 359 is divided by 11? (a) 8 (b) 4 (c) 7 (d) 6
  5. 5.What is the remainder when 176 is divided by 3? (a) 2 (b) 0 (c) 1
  6. 6.What is the smallest number you must add to 464 to make it divisible by 6? (a) 3 (b) 4 (c) 2 (d) 5
  7. 7.What is the smallest number you must add to 314 to make it divisible by 4? (a) 1 (b) 2 (c) 3
  8. 8.What is the remainder when 111 is divided by 11? (a) 1 (b) 0 (c) 2 (d) 10
  9. 9.What is the smallest number you must add to 388 to make it divisible by 5? (a) 3 (b) 1 (c) 2 (d) 4
  10. 10.Which of these numbers is divisible by 6? (a) 1289 (b) 1290 (c) 1298 (d) 1293
  11. 11.What is the remainder when 517 is divided by 7? (a) 5 (b) 1 (c) 4 (d) 6
  12. 12.What is the remainder when 713 is divided by 12? (a) 6 (b) 7 (c) 4 (d) 5
  13. 13.Which of these numbers is divisible by 8? (a) 5341 (b) 5328 (c) 5335 (d) 5321
  14. 14.Which of these numbers is divisible by 9? (a) 2316 (b) 2307 (c) 2304 (d) 2303
  15. 15.Which of these numbers is divisible by 11? (a) 3405 (b) 3417 (c) 3427 (d) 3410
  16. 16.Which of these numbers is divisible by 4? (a) 2986 (b) 2982 (c) 2991 (d) 2984
  17. 17.What is the remainder when 413 is divided by 8? (a) 3 (b) 5 (c) 6 (d) 4
  18. 18.What is the remainder when 431 is divided by 6? (a) 1 (b) 3 (c) 4 (d) 5
  19. 19.What is the smallest number you must add to 684 to make it divisible by 11? (a) 10 (b) 8 (c) 9 (d) 2
  20. 20.What is the smallest number you must add to 450 to make it divisible by 7? (a) 6 (b) 2 (c) 4 (d) 5
  21. 21.What is the smallest number you must add to 358 to make it divisible by 6? (a) 3 (b) 4 (c) 2 (d) 1
  22. 22.What is the units digit of 3^24? (a) 3 (b) 1 (c) 9 (d) 7
  23. 23.Which of these numbers is divisible by 3? (a) 1353 (b) 1357 (c) 1355 (d) 1351
  24. 24.Which of these numbers is divisible by 3? (a) 243 (b) 247 (c) 241 (d) 245
  25. 25.What is the remainder when 51 is divided by 4? (a) 0 (b) 1 (c) 3 (d) 2
  26. 26.Which of these numbers is divisible by 4? (a) 3249 (b) 3242 (c) 3244 (d) 3245
  27. 27.What is the smallest number you must add to 437 to make it divisible by 9? (a) 6 (b) 3 (c) 5 (d) 4
  28. 28.Which of these numbers is divisible by 8? (a) 4686 (b) 4672 (c) 4667 (d) 4677
  29. 29.Which of these numbers is divisible by 11? (a) 4924 (b) 4917 (c) 4915 (d) 4937
  30. 30.What is the smallest number you must add to 214 to make it divisible by 6? (a) 2 (b) 4 (c) 1 (d) 3
  31. 31.What is the smallest number you must add to 191 to make it divisible by 7? (a) 6 (b) 4 (c) 2 (d) 5
  32. 32.Which of these numbers is divisible by 8? (a) 4450 (b) 4444 (c) 4458 (d) 4448
  33. 33.Which of these numbers is divisible by 8? (a) 971 (b) 967 (c) 960 (d) 955
  34. 34.Which of these numbers is divisible by 6? (a) 4373 (b) 4383 (c) 4375 (d) 4374
  35. 35.What is the smallest number you must add to 820 to make it divisible by 12? (a) 4 (b) 8 (c) 9 (d) 7
  36. 36.Which of these numbers is divisible by 4? (a) 2049 (b) 2054 (c) 2057 (d) 2052
  37. 37.What is the units digit of 2^42? (a) 8 (b) 2 (c) 6 (d) 4
  38. 38.What is the smallest number you must add to 235 to make it divisible by 6? (a) 4 (b) 3 (c) 5 (d) 1
  39. 39.What is the units digit of 9^46? (a) 1 (b) 2 (c) 9 (d) 0
  40. 40.What is the smallest number you must add to 834 to make it divisible by 12? (a) 7 (b) 6 (c) 5 (d) 8

Answer key

  1. 504
  2. 7
  3. 4
  4. 7
  5. 2
  6. 4
  7. 2
  8. 1
  9. 2
  10. 1290
  11. 6
  12. 5
  13. 5328
  14. 2304
  15. 3410
  16. 2984
  17. 5
  18. 5
  19. 9
  20. 5
  21. 2
  22. 1
  23. 1353
  24. 243
  25. 3
  26. 3244
  27. 4
  28. 4672
  29. 4917
  30. 2
  31. 5
  32. 4448
  33. 960
  34. 4374
  35. 8
  36. 2052
  37. 4
  38. 5
  39. 1
  40. 6

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