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Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 245 · 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 91 is divided by 4? (a) 2 (b) 1 (c) 0 (d) 3
  2. 2.What is the smallest number you must add to 658 to make it divisible by 10? (a) 3 (b) 8 (c) 1 (d) 2
  3. 3.What is the smallest number you must add to 61 to make it divisible by 3? (a) 2 (b) 1
  4. 4.What is the units digit of 7^26? (a) 7 (b) 1 (c) 9 (d) 3
  5. 5.What is the units digit of 2^34? (a) 6 (b) 8 (c) 4 (d) 2
  6. 6.What is the smallest number you must add to 65 to make it divisible by 6? (a) 2 (b) 5 (c) 1 (d) 3
  7. 7.Which of these numbers is divisible by 4? (a) 2205 (b) 2202 (c) 2200 (d) 2198
  8. 8.Which of these numbers is divisible by 4? (a) 244 (b) 242 (c) 251 (d) 247
  9. 9.What is the units digit of 7^50? (a) 3 (b) 7 (c) 1 (d) 9
  10. 10.What is the remainder when 193 is divided by 4? (a) 0 (b) 1 (c) 2 (d) 3
  11. 11.What is the remainder when 32 is divided by 3? (a) 0 (b) 2 (c) 1
  12. 12.What is the smallest number you must add to 379 to make it divisible by 6? (a) 5 (b) 1 (c) 4 (d) 3
  13. 13.Which of these numbers is divisible by 9? (a) 8075 (b) 8057 (c) 8072 (d) 8064
  14. 14.What is the remainder when 109 is divided by 7? (a) 4 (b) 3 (c) 6 (d) 5
  15. 15.What is the smallest number you must add to 193 to make it divisible by 4? (a) 1 (b) 2 (c) 3
  16. 16.What is the smallest number you must add to 873 to make it divisible by 12? (a) 4 (b) 2 (c) 9 (d) 3
  17. 17.What is the units digit of 7^57? (a) 9 (b) 1 (c) 7 (d) 3
  18. 18.Which of these numbers is divisible by 3? (a) 536 (b) 534 (c) 539 (d) 533
  19. 19.Which of these numbers is divisible by 4? (a) 2893 (b) 2897 (c) 2903 (d) 2896
  20. 20.Which of these numbers is divisible by 3? (a) 845 (b) 850 (c) 846 (d) 847
  21. 21.What is the remainder when 350 is divided by 6? (a) 1 (b) 4 (c) 3 (d) 2
  22. 22.What is the smallest number you must add to 162 to make it divisible by 7? (a) 1 (b) 4 (c) 5 (d) 6
  23. 23.What is the remainder when 377 is divided by 6? (a) 4 (b) 1 (c) 5 (d) 3
  24. 24.What is the remainder when 208 is divided by 5? (a) 3 (b) 4 (c) 2 (d) 1
  25. 25.What is the smallest number you must add to 141 to make it divisible by 6? (a) 2 (b) 3 (c) 5 (d) 4
  26. 26.Which of these numbers is divisible by 9? (a) 2741 (b) 2729 (c) 2736 (d) 2748
  27. 27.What is the smallest number you must add to 429 to make it divisible by 6? (a) 4 (b) 3 (c) 2 (d) 5
  28. 28.Which of these numbers is divisible by 3? (a) 1432 (b) 1428 (c) 1430 (d) 1426
  29. 29.What is the remainder when 138 is divided by 9? (a) 4 (b) 2 (c) 3 (d) 6
  30. 30.What is the smallest number you must add to 421 to make it divisible by 6? (a) 1 (b) 4 (c) 3 (d) 5
  31. 31.What is the smallest number you must add to 352 to make it divisible by 6? (a) 4 (b) 3 (c) 2 (d) 1
  32. 32.What is the units digit of 9^54? (a) 0 (b) 9 (c) 2 (d) 1
  33. 33.What is the smallest number you must add to 194 to make it divisible by 4? (a) 2 (b) 1 (c) 3
  34. 34.What is the units digit of 3^22? (a) 3 (b) 7 (c) 1 (d) 9
  35. 35.What is the units digit of 9^33? (a) 8 (b) 1 (c) 7 (d) 9
  36. 36.What is the smallest number you must add to 128 to make it divisible by 5? (a) 3 (b) 2 (c) 4 (d) 1
  37. 37.What is the smallest number you must add to 258 to make it divisible by 12? (a) 8 (b) 7 (c) 5 (d) 6
  38. 38.Which of these numbers is divisible by 8? (a) 3356 (b) 3361 (c) 3350 (d) 3352
  39. 39.Which of these numbers is divisible by 4? (a) 584 (b) 585 (c) 591 (d) 583
  40. 40.What is the smallest number you must add to 556 to make it divisible by 9? (a) 7 (b) 3 (c) 1 (d) 2

Answer key

  1. 3
  2. 2
  3. 2
  4. 9
  5. 4
  6. 1
  7. 2200
  8. 244
  9. 9
  10. 1
  11. 2
  12. 5
  13. 8064
  14. 4
  15. 3
  16. 3
  17. 7
  18. 534
  19. 2896
  20. 846
  21. 2
  22. 6
  23. 5
  24. 3
  25. 3
  26. 2736
  27. 3
  28. 1428
  29. 3
  30. 5
  31. 2
  32. 1
  33. 2
  34. 9
  35. 9
  36. 2
  37. 6
  38. 3352
  39. 584
  40. 2

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