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Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 14 · 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 343 to make it divisible by 5? (a) 1 (b) 4 (c) 3 (d) 2
  2. 2.What is the remainder when 279 is divided by 6? (a) 4 (b) 3 (c) 5 (d) 2
  3. 3.What is the units digit of 9^43? (a) 8 (b) 7 (c) 9 (d) 1
  4. 4.What is the remainder when 455 is divided by 6? (a) 1 (b) 5 (c) 3 (d) 4
  5. 5.Which of these numbers is divisible by 3? (a) 1280 (b) 1276 (c) 1273 (d) 1275
  6. 6.Which of these numbers is divisible by 11? (a) 819 (b) 803 (c) 800 (d) 812
  7. 7.What is the units digit of 9^35? (a) 9 (b) 1 (c) 7 (d) 8
  8. 8.Which of these numbers is divisible by 9? (a) 7646 (b) 7641 (c) 7654 (d) 7640
  9. 9.What is the remainder when 469 is divided by 11? (a) 7 (b) 4 (c) 8 (d) 6
  10. 10.What is the smallest number you must add to 45 to make it divisible by 4? (a) 1 (b) 2 (c) 3
  11. 11.What is the smallest number you must add to 242 to make it divisible by 4? (a) 2 (b) 1 (c) 3
  12. 12.What is the remainder when 139 is divided by 6? (a) 5 (b) 1 (c) 2 (d) 0
  13. 13.Which of these numbers is divisible by 8? (a) 834 (b) 820 (c) 824 (d) 825
  14. 14.What is the smallest number you must add to 293 to make it divisible by 10? (a) 3 (b) 6 (c) 8 (d) 7
  15. 15.What is the smallest number you must add to 176 to make it divisible by 3? (a) 1 (b) 2
  16. 16.What is the remainder when 337 is divided by 6? (a) 2 (b) 0 (c) 1 (d) 5
  17. 17.Which of these numbers is divisible by 9? (a) 1144 (b) 1138 (c) 1134 (d) 1127
  18. 18.What is the smallest number you must add to 425 to make it divisible by 6? (a) 2 (b) 1 (c) 5 (d) 3
  19. 19.Which of these numbers is divisible by 6? (a) 1949 (b) 1951 (c) 1950 (d) 1959
  20. 20.What is the smallest number you must add to 801 to make it divisible by 11? (a) 1 (b) 9 (c) 2 (d) 3
  21. 21.Which of these numbers is divisible by 9? (a) 1445 (b) 1449 (c) 1455 (d) 1460
  22. 22.Which of these numbers is divisible by 11? (a) 9825 (b) 9834 (c) 9837 (d) 9846
  23. 23.Which of these numbers is divisible by 4? (a) 1553 (b) 1561 (c) 1559 (d) 1556
  24. 24.Which of these numbers is divisible by 6? (a) 4084 (b) 4088 (c) 4086 (d) 4093
  25. 25.Which of these numbers is divisible by 9? (a) 2405 (b) 2403 (c) 2417 (d) 2400
  26. 26.What is the remainder when 608 is divided by 10? (a) 8 (b) 9 (c) 7 (d) 2
  27. 27.What is the smallest number you must add to 136 to make it divisible by 3? (a) 2 (b) 1
  28. 28.What is the units digit of 4^41? (a) 6 (b) 5 (c) 4 (d) 3
  29. 29.Which of these numbers is divisible by 6? (a) 1253 (b) 1254 (c) 1261 (d) 1259
  30. 30.What is the units digit of 7^25? (a) 9 (b) 3 (c) 1 (d) 7
  31. 31.What is the smallest number you must add to 285 to make it divisible by 4? (a) 1 (b) 3 (c) 2
  32. 32.What is the remainder when 167 is divided by 10? (a) 6 (b) 8 (c) 3 (d) 7
  33. 33.What is the smallest number you must add to 138 to make it divisible by 5? (a) 4 (b) 3 (c) 2 (d) 1
  34. 34.What is the smallest number you must add to 334 to make it divisible by 5? (a) 3 (b) 1 (c) 4 (d) 2
  35. 35.What is the smallest number you must add to 809 to make it divisible by 12? (a) 8 (b) 6 (c) 7 (d) 5
  36. 36.What is the units digit of 8^51? (a) 2 (b) 8 (c) 6 (d) 4
  37. 37.What is the units digit of 9^45? (a) 7 (b) 1 (c) 8 (d) 9
  38. 38.What is the smallest number you must add to 165 to make it divisible by 7? (a) 2 (b) 3 (c) 4 (d) 5
  39. 39.Which of these numbers is divisible by 4? (a) 2046 (b) 2040 (c) 2039 (d) 2041
  40. 40.What is the units digit of 7^48? (a) 7 (b) 9 (c) 3 (d) 1

Answer key

  1. 2
  2. 3
  3. 9
  4. 5
  5. 1275
  6. 803
  7. 9
  8. 7641
  9. 7
  10. 3
  11. 2
  12. 1
  13. 824
  14. 7
  15. 1
  16. 1
  17. 1134
  18. 1
  19. 1950
  20. 2
  21. 1449
  22. 9834
  23. 1556
  24. 4086
  25. 2403
  26. 8
  27. 2
  28. 4
  29. 1254
  30. 7
  31. 3
  32. 7
  33. 2
  34. 1
  35. 7
  36. 2
  37. 9
  38. 3
  39. 2040
  40. 1

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