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Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 170 · 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 units digit of 8^14? (a) 4 (b) 8 (c) 2 (d) 6
  2. 2.Which of these numbers is divisible by 4? (a) 1346 (b) 1339 (c) 1340 (d) 1342
  3. 3.What is the units digit of 3^46? (a) 9 (b) 1 (c) 3 (d) 7
  4. 4.What is the smallest number you must add to 508 to make it divisible by 10? (a) 8 (b) 1 (c) 3 (d) 2
  5. 5.What is the smallest number you must add to 270 to make it divisible by 4? (a) 3 (b) 2 (c) 1
  6. 6.What is the remainder when 717 is divided by 11? (a) 3 (b) 9 (c) 1 (d) 2
  7. 7.What is the units digit of 7^54? (a) 3 (b) 1 (c) 7 (d) 9
  8. 8.What is the smallest number you must add to 202 to make it divisible by 4? (a) 3 (b) 1 (c) 2
  9. 9.What is the smallest number you must add to 596 to make it divisible by 10? (a) 6 (b) 3 (c) 4 (d) 5
  10. 10.What is the remainder when 121 is divided by 12? (a) 2 (b) 1 (c) 11 (d) 0
  11. 11.What is the smallest number you must add to 359 to make it divisible by 6? (a) 3 (b) 2 (c) 5 (d) 1
  12. 12.What is the smallest number you must add to 116 to make it divisible by 9? (a) 3 (b) 2 (c) 8 (d) 1
  13. 13.What is the remainder when 287 is divided by 6? (a) 5 (b) 3 (c) 4 (d) 1
  14. 14.Which of these numbers is divisible by 8? (a) 340 (b) 336 (c) 351 (d) 330
  15. 15.What is the remainder when 488 is divided by 7? (a) 4 (b) 6 (c) 2 (d) 5
  16. 16.What is the remainder when 322 is divided by 8? (a) 3 (b) 2 (c) 1 (d) 6
  17. 17.What is the smallest number you must add to 61 to make it divisible by 6? (a) 4 (b) 5 (c) 3 (d) 1
  18. 18.What is the remainder when 89 is divided by 4? (a) 2 (b) 3 (c) 1 (d) 0
  19. 19.Which of these numbers is divisible by 4? (a) 855 (b) 850 (c) 852 (d) 859
  20. 20.Which of these numbers is divisible by 4? (a) 2504 (b) 2507 (c) 2501 (d) 2511
  21. 21.Which of these numbers is divisible by 11? (a) 8875 (b) 8877 (c) 8883 (d) 8889
  22. 22.Which of these numbers is divisible by 4? (a) 2514 (b) 2512 (c) 2511 (d) 2519
  23. 23.What is the units digit of 7^36? (a) 1 (b) 7 (c) 9 (d) 3
  24. 24.What is the remainder when 111 is divided by 4? (a) 3 (b) 1 (c) 0 (d) 2
  25. 25.What is the units digit of 8^15? (a) 8 (b) 6 (c) 4 (d) 2
  26. 26.Which of these numbers is divisible by 8? (a) 2811 (b) 2822 (c) 2826 (d) 2816
  27. 27.What is the remainder when 243 is divided by 4? (a) 3 (b) 2 (c) 1 (d) 0
  28. 28.What is the smallest number you must add to 515 to make it divisible by 11? (a) 2 (b) 9 (c) 1 (d) 3
  29. 29.Which of these numbers is divisible by 6? (a) 3906 (b) 3917 (c) 3911 (d) 3901
  30. 30.What is the smallest number you must add to 167 to make it divisible by 4? (a) 3 (b) 1 (c) 2
  31. 31.What is the smallest number you must add to 481 to make it divisible by 6? (a) 1 (b) 4 (c) 5 (d) 3
  32. 32.What is the smallest number you must add to 266 to make it divisible by 5? (a) 1 (b) 4 (c) 3 (d) 2
  33. 33.What is the units digit of 8^25? (a) 4 (b) 8 (c) 2 (d) 6
  34. 34.Which of these numbers is divisible by 4? (a) 1983 (b) 1974 (c) 1976 (d) 1978
  35. 35.What is the units digit of 7^16? (a) 7 (b) 1 (c) 9 (d) 3
  36. 36.What is the smallest number you must add to 114 to make it divisible by 8? (a) 2 (b) 5 (c) 6 (d) 7
  37. 37.What is the smallest number you must add to 161 to make it divisible by 5? (a) 1 (b) 2 (c) 4 (d) 3
  38. 38.Which of these numbers is divisible by 9? (a) 5468 (b) 5488 (c) 5477 (d) 5472
  39. 39.What is the units digit of 3^51? (a) 7 (b) 1 (c) 9 (d) 3
  40. 40.What is the smallest number you must add to 289 to make it divisible by 4? (a) 1 (b) 2 (c) 3

Answer key

  1. 4
  2. 1340
  3. 9
  4. 2
  5. 2
  6. 2
  7. 9
  8. 2
  9. 4
  10. 1
  11. 1
  12. 1
  13. 5
  14. 336
  15. 5
  16. 2
  17. 5
  18. 1
  19. 852
  20. 2504
  21. 8877
  22. 2512
  23. 1
  24. 3
  25. 2
  26. 2816
  27. 3
  28. 2
  29. 3906
  30. 1
  31. 5
  32. 4
  33. 8
  34. 1976
  35. 1
  36. 6
  37. 4
  38. 5472
  39. 7
  40. 3

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