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Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 768 · 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 53 is divided by 4? (a) 1 (b) 2 (c) 3 (d) 0
  2. 2.What is the smallest number you must add to 197 to make it divisible by 9? (a) 1 (b) 8 (c) 2 (d) 3
  3. 3.What is the units digit of 8^20? (a) 8 (b) 6 (c) 2 (d) 4
  4. 4.What is the remainder when 362 is divided by 7? (a) 4 (b) 5 (c) 2 (d) 6
  5. 5.What is the units digit of 3^49? (a) 9 (b) 7 (c) 1 (d) 3
  6. 6.What is the smallest number you must add to 62 to make it divisible by 6? (a) 3 (b) 5 (c) 2 (d) 4
  7. 7.What is the units digit of 4^50? (a) 6 (b) 5 (c) 7 (d) 4
  8. 8.What is the smallest number you must add to 237 to make it divisible by 10? (a) 7 (b) 4 (c) 2 (d) 3
  9. 9.What is the smallest number you must add to 43 to make it divisible by 4? (a) 2 (b) 3 (c) 1
  10. 10.Which of these numbers is divisible by 3? (a) 187 (b) 190 (c) 184 (d) 186
  11. 11.Which of these numbers is divisible by 11? (a) 2036 (b) 2035 (c) 2053 (d) 2028
  12. 12.Which of these numbers is divisible by 4? (a) 2294 (b) 2297 (c) 2302 (d) 2296
  13. 13.What is the units digit of 2^48? (a) 6 (b) 8 (c) 4 (d) 2
  14. 14.What is the smallest number you must add to 605 to make it divisible by 9? (a) 7 (b) 2 (c) 8 (d) 6
  15. 15.What is the smallest number you must add to 219 to make it divisible by 7? (a) 4 (b) 5 (c) 6 (d) 2
  16. 16.What is the remainder when 589 is divided by 12? (a) 2 (b) 1 (c) 11 (d) 0
  17. 17.What is the remainder when 167 is divided by 10? (a) 3 (b) 7 (c) 6 (d) 8
  18. 18.What is the remainder when 101 is divided by 4? (a) 3 (b) 0 (c) 2 (d) 1
  19. 19.What is the smallest number you must add to 159 to make it divisible by 5? (a) 1 (b) 3 (c) 2 (d) 4
  20. 20.Which of these numbers is divisible by 4? (a) 3511 (b) 3514 (c) 3506 (d) 3508
  21. 21.What is the smallest number you must add to 407 to make it divisible by 9? (a) 6 (b) 2 (c) 8 (d) 7
  22. 22.What is the units digit of 9^24? (a) 0 (b) 9 (c) 1 (d) 2
  23. 23.What is the units digit of 2^51? (a) 8 (b) 4 (c) 2 (d) 6
  24. 24.Which of these numbers is divisible by 11? (a) 606 (b) 602 (c) 591 (d) 594
  25. 25.What is the units digit of 4^52? (a) 4 (b) 7 (c) 5 (d) 6
  26. 26.What is the units digit of 3^38? (a) 1 (b) 3 (c) 7 (d) 9
  27. 27.Which of these numbers is divisible by 9? (a) 1613 (b) 1610 (c) 1621 (d) 1611
  28. 28.Which of these numbers is divisible by 9? (a) 754 (b) 759 (c) 745 (d) 747
  29. 29.What is the remainder when 127 is divided by 6? (a) 5 (b) 0 (c) 2 (d) 1
  30. 30.Which of these numbers is divisible by 9? (a) 7499 (b) 7492 (c) 7513 (d) 7497
  31. 31.What is the units digit of 9^12? (a) 9 (b) 2 (c) 0 (d) 1
  32. 32.What is the remainder when 639 is divided by 11? (a) 1 (b) 10 (c) 2 (d) 0
  33. 33.What is the remainder when 309 is divided by 7? (a) 0 (b) 2 (c) 1 (d) 6
  34. 34.What is the smallest number you must add to 101 to make it divisible by 5? (a) 2 (b) 4 (c) 3 (d) 1
  35. 35.What is the units digit of 3^18? (a) 3 (b) 9 (c) 1 (d) 7
  36. 36.Which of these numbers is divisible by 4? (a) 285 (b) 282 (c) 291 (d) 284
  37. 37.What is the remainder when 107 is divided by 3? (a) 1 (b) 2 (c) 0
  38. 38.What is the units digit of 3^46? (a) 9 (b) 3 (c) 1 (d) 7
  39. 39.What is the remainder when 112 is divided by 9? (a) 5 (b) 6 (c) 3 (d) 4
  40. 40.What is the smallest number you must add to 411 to make it divisible by 7? (a) 2 (b) 1 (c) 3 (d) 5

Answer key

  1. 1
  2. 1
  3. 6
  4. 5
  5. 3
  6. 4
  7. 6
  8. 3
  9. 1
  10. 186
  11. 2035
  12. 2296
  13. 6
  14. 7
  15. 5
  16. 1
  17. 7
  18. 1
  19. 1
  20. 3508
  21. 7
  22. 1
  23. 8
  24. 594
  25. 6
  26. 9
  27. 1611
  28. 747
  29. 1
  30. 7497
  31. 1
  32. 1
  33. 1
  34. 4
  35. 9
  36. 284
  37. 2
  38. 9
  39. 4
  40. 2

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