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Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 502 · 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 4^23? (a) 6 (b) 3 (c) 4 (d) 5
  2. 2.What is the smallest number you must add to 755 to make it divisible by 11? (a) 7 (b) 5 (c) 4 (d) 3
  3. 3.Which of these numbers is divisible by 11? (a) 2894 (b) 2890 (c) 2893 (d) 2908
  4. 4.What is the smallest number you must add to 263 to make it divisible by 6? (a) 2 (b) 1 (c) 3 (d) 5
  5. 5.What is the smallest number you must add to 385 to make it divisible by 10? (a) 5 (b) 4 (c) 6 (d) 7
  6. 6.What is the units digit of 7^39? (a) 9 (b) 1 (c) 3 (d) 7
  7. 7.What is the units digit of 4^10? (a) 5 (b) 7 (c) 6 (d) 4
  8. 8.What is the smallest number you must add to 104 to make it divisible by 5? (a) 1 (b) 2 (c) 4 (d) 3
  9. 9.What is the units digit of 2^32? (a) 6 (b) 8 (c) 2 (d) 4
  10. 10.Which of these numbers is divisible by 4? (a) 2143 (b) 2149 (c) 2145 (d) 2144
  11. 11.What is the remainder when 490 is divided by 8? (a) 6 (b) 2 (c) 3 (d) 1
  12. 12.What is the remainder when 290 is divided by 4? (a) 0 (b) 1 (c) 3 (d) 2
  13. 13.Which of these numbers is divisible by 3? (a) 2151 (b) 2153 (c) 2149 (d) 2155
  14. 14.What is the smallest number you must add to 678 to make it divisible by 9? (a) 5 (b) 7 (c) 3 (d) 6
  15. 15.What is the smallest number you must add to 472 to make it divisible by 9? (a) 4 (b) 5 (c) 7 (d) 6
  16. 16.What is the remainder when 176 is divided by 3? (a) 0 (b) 1 (c) 2
  17. 17.What is the smallest number you must add to 193 to make it divisible by 3? (a) 2 (b) 1
  18. 18.What is the remainder when 73 is divided by 4? (a) 3 (b) 0 (c) 2 (d) 1
  19. 19.What is the units digit of 8^51? (a) 2 (b) 6 (c) 8 (d) 4
  20. 20.What is the remainder when 575 is divided by 12? (a) 9 (b) 10 (c) 11 (d) 1
  21. 21.Which of these numbers is divisible by 4? (a) 2183 (b) 2180 (c) 2185 (d) 2178
  22. 22.What is the smallest number you must add to 412 to make it divisible by 12? (a) 8 (b) 7 (c) 9 (d) 4
  23. 23.Which of these numbers is divisible by 8? (a) 3141 (b) 3131 (c) 3128 (d) 3124
  24. 24.Which of these numbers is divisible by 6? (a) 1003 (b) 996 (c) 994 (d) 999
  25. 25.Which of these numbers is divisible by 3? (a) 420 (b) 421 (c) 419 (d) 425
  26. 26.What is the remainder when 185 is divided by 7? (a) 2 (b) 5 (c) 4 (d) 3
  27. 27.What is the remainder when 275 is divided by 6? (a) 4 (b) 3 (c) 1 (d) 5
  28. 28.What is the units digit of 4^31? (a) 4 (b) 5 (c) 3 (d) 6
  29. 29.What is the smallest number you must add to 286 to make it divisible by 5? (a) 3 (b) 4 (c) 2 (d) 1
  30. 30.What is the smallest number you must add to 122 to make it divisible by 3? (a) 2 (b) 1
  31. 31.What is the remainder when 353 is divided by 8? (a) 7 (b) 2 (c) 1 (d) 0
  32. 32.Which of these numbers is divisible by 11? (a) 5080 (b) 5099 (c) 5082 (d) 5090
  33. 33.Which of these numbers is divisible by 11? (a) 8974 (b) 8954 (c) 8949 (d) 8957
  34. 34.Which of these numbers is divisible by 8? (a) 2874 (b) 2860 (c) 2864 (d) 2870
  35. 35.What is the remainder when 97 is divided by 9? (a) 2 (b) 7 (c) 8 (d) 6
  36. 36.What is the smallest number you must add to 197 to make it divisible by 4? (a) 1 (b) 2 (c) 3
  37. 37.What is the smallest number you must add to 200 to make it divisible by 6? (a) 3 (b) 4 (c) 2 (d) 5
  38. 38.What is the remainder when 125 is divided by 4? (a) 2 (b) 1 (c) 3 (d) 0
  39. 39.What is the smallest number you must add to 169 to make it divisible by 9? (a) 1 (b) 3 (c) 7 (d) 2
  40. 40.What is the remainder when 373 is divided by 8? (a) 4 (b) 5 (c) 3 (d) 6

Answer key

  1. 4
  2. 4
  3. 2893
  4. 1
  5. 5
  6. 3
  7. 6
  8. 1
  9. 6
  10. 2144
  11. 2
  12. 2
  13. 2151
  14. 6
  15. 5
  16. 2
  17. 2
  18. 1
  19. 2
  20. 11
  21. 2180
  22. 8
  23. 3128
  24. 996
  25. 420
  26. 3
  27. 5
  28. 4
  29. 4
  30. 1
  31. 1
  32. 5082
  33. 8954
  34. 2864
  35. 7
  36. 3
  37. 4
  38. 1
  39. 2
  40. 5

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