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Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 219 · 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 remainder when 265 is divided by 7? (a) 5 (b) 1 (c) 4 (d) 6
  2. 2.Which of these numbers is divisible by 6? (a) 5018 (b) 5016 (c) 5013 (d) 5024
  3. 3.What is the smallest number you must add to 242 to make it divisible by 6? (a) 5 (b) 2 (c) 3 (d) 4
  4. 4.Which of these numbers is divisible by 3? (a) 1042 (b) 1048 (c) 1044 (d) 1045
  5. 5.Which of these numbers is divisible by 11? (a) 1619 (b) 1634 (c) 1617 (d) 1614
  6. 6.What is the remainder when 117 is divided by 4? (a) 3 (b) 1 (c) 2 (d) 0
  7. 7.What is the smallest number you must add to 282 to make it divisible by 7? (a) 5 (b) 6 (c) 2 (d) 4
  8. 8.What is the smallest number you must add to 397 to make it divisible by 7? (a) 1 (b) 5 (c) 2 (d) 3
  9. 9.Which of these numbers is divisible by 4? (a) 1306 (b) 1311 (c) 1313 (d) 1308
  10. 10.Which of these numbers is divisible by 4? (a) 2820 (b) 2818 (c) 2822 (d) 2825
  11. 11.Which of these numbers is divisible by 4? (a) 2333 (b) 2339 (c) 2330 (d) 2332
  12. 12.What is the units digit of 3^18? (a) 3 (b) 9 (c) 7 (d) 1
  13. 13.What is the smallest number you must add to 233 to make it divisible by 10? (a) 6 (b) 7 (c) 3 (d) 8
  14. 14.What is the smallest number you must add to 662 to make it divisible by 9? (a) 4 (b) 5 (c) 6 (d) 3
  15. 15.What is the smallest number you must add to 635 to make it divisible by 9? (a) 6 (b) 4 (c) 3 (d) 5
  16. 16.What is the remainder when 331 is divided by 12? (a) 6 (b) 5 (c) 7 (d) 8
  17. 17.What is the units digit of 7^48? (a) 7 (b) 3 (c) 9 (d) 1
  18. 18.What is the smallest number you must add to 706 to make it divisible by 9? (a) 4 (b) 6 (c) 5 (d) 7
  19. 19.What is the units digit of 9^51? (a) 1 (b) 7 (c) 8 (d) 9
  20. 20.What is the smallest number you must add to 237 to make it divisible by 5? (a) 1 (b) 3 (c) 2 (d) 4
  21. 21.Which of these numbers is divisible by 8? (a) 1170 (b) 1160 (c) 1155 (d) 1166
  22. 22.Which of these numbers is divisible by 9? (a) 1027 (b) 1022 (c) 1017 (d) 1009
  23. 23.What is the units digit of 9^43? (a) 8 (b) 1 (c) 9 (d) 7
  24. 24.What is the smallest number you must add to 545 to make it divisible by 8? (a) 5 (b) 6 (c) 7 (d) 1
  25. 25.What is the remainder when 104 is divided by 5? (a) 1 (b) 3 (c) 2 (d) 4
  26. 26.Which of these numbers is divisible by 4? (a) 2579 (b) 2581 (c) 2580 (d) 2585
  27. 27.Which of these numbers is divisible by 8? (a) 5869 (b) 5881 (c) 5879 (d) 5872
  28. 28.Which of these numbers is divisible by 4? (a) 1326 (b) 1330 (c) 1328 (d) 1333
  29. 29.What is the smallest number you must add to 86 to make it divisible by 8? (a) 3 (b) 1 (c) 6 (d) 2
  30. 30.What is the smallest number you must add to 295 to make it divisible by 10? (a) 4 (b) 5 (c) 7 (d) 6
  31. 31.Which of these numbers is divisible by 11? (a) 844 (b) 856 (c) 836 (d) 829
  32. 32.What is the units digit of 9^52? (a) 0 (b) 1 (c) 2 (d) 9
  33. 33.What is the units digit of 9^54? (a) 9 (b) 0 (c) 1 (d) 2
  34. 34.Which of these numbers is divisible by 3? (a) 316 (b) 320 (c) 313 (d) 315
  35. 35.What is the remainder when 129 is divided by 9? (a) 2 (b) 3 (c) 4 (d) 6
  36. 36.What is the remainder when 220 is divided by 9? (a) 5 (b) 3 (c) 6 (d) 4
  37. 37.What is the units digit of 4^58? (a) 7 (b) 5 (c) 6 (d) 4
  38. 38.What is the remainder when 298 is divided by 9? (a) 2 (b) 0 (c) 8 (d) 1
  39. 39.What is the remainder when 170 is divided by 12? (a) 10 (b) 2 (c) 1 (d) 3
  40. 40.What is the smallest number you must add to 373 to make it divisible by 6? (a) 4 (b) 1 (c) 3 (d) 5

Answer key

  1. 6
  2. 5016
  3. 4
  4. 1044
  5. 1617
  6. 1
  7. 5
  8. 2
  9. 1308
  10. 2820
  11. 2332
  12. 9
  13. 7
  14. 4
  15. 4
  16. 7
  17. 1
  18. 5
  19. 9
  20. 3
  21. 1160
  22. 1017
  23. 9
  24. 7
  25. 4
  26. 2580
  27. 5872
  28. 1328
  29. 2
  30. 5
  31. 836
  32. 1
  33. 1
  34. 315
  35. 3
  36. 4
  37. 6
  38. 1
  39. 2
  40. 5

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