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Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 620 · 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 smallest number you must add to 353 to make it divisible by 11? (a) 10 (b) 8 (c) 9 (d) 1
  2. 2.What is the remainder when 455 is divided by 9? (a) 7 (b) 4 (c) 5 (d) 6
  3. 3.What is the smallest number you must add to 270 to make it divisible by 7? (a) 5 (b) 3 (c) 4 (d) 2
  4. 4.What is the smallest number you must add to 172 to make it divisible by 6? (a) 3 (b) 4 (c) 2 (d) 1
  5. 5.What is the smallest number you must add to 264 to make it divisible by 5? (a) 2 (b) 3 (c) 4 (d) 1
  6. 6.What is the remainder when 138 is divided by 9? (a) 6 (b) 4 (c) 2 (d) 3
  7. 7.What is the smallest number you must add to 248 to make it divisible by 6? (a) 5 (b) 4 (c) 3 (d) 2
  8. 8.What is the remainder when 254 is divided by 6? (a) 3 (b) 1 (c) 2 (d) 4
  9. 9.Which of these numbers is divisible by 3? (a) 2071 (b) 2066 (c) 2068 (d) 2067
  10. 10.What is the remainder when 672 is divided by 10? (a) 3 (b) 1 (c) 2 (d) 8
  11. 11.Which of these numbers is divisible by 4? (a) 2198 (b) 2205 (c) 2202 (d) 2200
  12. 12.What is the smallest number you must add to 116 to make it divisible by 11? (a) 4 (b) 7 (c) 5 (d) 6
  13. 13.What is the units digit of 3^10? (a) 9 (b) 3 (c) 7 (d) 1
  14. 14.What is the smallest number you must add to 246 to make it divisible by 10? (a) 5 (b) 4 (c) 6 (d) 3
  15. 15.Which of these numbers is divisible by 8? (a) 6577 (b) 6585 (c) 6597 (d) 6584
  16. 16.Which of these numbers is divisible by 6? (a) 5122 (b) 5115 (c) 5118 (d) 5129
  17. 17.What is the units digit of 7^10? (a) 3 (b) 7 (c) 9 (d) 1
  18. 18.Which of these numbers is divisible by 11? (a) 1113 (b) 1111 (c) 1110 (d) 1130
  19. 19.What is the units digit of 3^59? (a) 9 (b) 3 (c) 1 (d) 7
  20. 20.What is the smallest number you must add to 465 to make it divisible by 7? (a) 6 (b) 4 (c) 3 (d) 5
  21. 21.What is the remainder when 933 is divided by 12? (a) 3 (b) 8 (c) 10 (d) 9
  22. 22.What is the smallest number you must add to 302 to make it divisible by 6? (a) 5 (b) 2 (c) 4 (d) 3
  23. 23.What is the remainder when 356 is divided by 6? (a) 4 (b) 1 (c) 3 (d) 2
  24. 24.Which of these numbers is divisible by 11? (a) 7769 (b) 7759 (c) 7780 (d) 7766
  25. 25.What is the remainder when 368 is divided by 5? (a) 3 (b) 4 (c) 1 (d) 2
  26. 26.What is the smallest number you must add to 179 to make it divisible by 3? (a) 1 (b) 2
  27. 27.What is the smallest number you must add to 91 to make it divisible by 5? (a) 3 (b) 4 (c) 2 (d) 1
  28. 28.Which of these numbers is divisible by 4? (a) 960 (b) 966 (c) 963 (d) 958
  29. 29.Which of these numbers is divisible by 8? (a) 3690 (b) 3702 (c) 3706 (d) 3696
  30. 30.What is the remainder when 342 is divided by 12? (a) 7 (b) 8 (c) 5 (d) 6
  31. 31.What is the remainder when 650 is divided by 12? (a) 1 (b) 3 (c) 2 (d) 10
  32. 32.Which of these numbers is divisible by 11? (a) 3208 (b) 3191 (c) 3220 (d) 3201
  33. 33.What is the smallest number you must add to 127 to make it divisible by 3? (a) 1 (b) 2
  34. 34.What is the units digit of 4^20? (a) 6 (b) 4 (c) 7 (d) 5
  35. 35.Which of these numbers is divisible by 3? (a) 2585 (b) 2579 (c) 2581 (d) 2580
  36. 36.What is the smallest number you must add to 129 to make it divisible by 5? (a) 3 (b) 1 (c) 4 (d) 2
  37. 37.Which of these numbers is divisible by 11? (a) 857 (b) 847 (c) 840 (d) 861
  38. 38.What is the units digit of 2^22? (a) 8 (b) 2 (c) 6 (d) 4
  39. 39.Which of these numbers is divisible by 6? (a) 1036 (b) 1027 (c) 1032 (d) 1041
  40. 40.Which of these numbers is divisible by 11? (a) 3173 (b) 3189 (c) 3168 (d) 3165

Answer key

  1. 10
  2. 5
  3. 3
  4. 2
  5. 1
  6. 3
  7. 4
  8. 2
  9. 2067
  10. 2
  11. 2200
  12. 5
  13. 9
  14. 4
  15. 6584
  16. 5118
  17. 9
  18. 1111
  19. 7
  20. 4
  21. 9
  22. 4
  23. 2
  24. 7766
  25. 3
  26. 1
  27. 4
  28. 960
  29. 3696
  30. 6
  31. 2
  32. 3201
  33. 2
  34. 6
  35. 2580
  36. 1
  37. 847
  38. 4
  39. 1032
  40. 3168

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