← Divisibility, remainders and units digits lesson New set →

Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 926 · 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 9^23? (a) 8 (b) 9 (c) 1 (d) 7
  2. 2.Which of these numbers is divisible by 3? (a) 141 (b) 145 (c) 140 (d) 142
  3. 3.What is the smallest number you must add to 44 to make it divisible by 3? (a) 2 (b) 1
  4. 4.Which of these numbers is divisible by 4? (a) 1816 (b) 1817 (c) 1815 (d) 1821
  5. 5.What is the smallest number you must add to 213 to make it divisible by 9? (a) 4 (b) 3 (c) 6 (d) 2
  6. 6.What is the remainder when 99 is divided by 5? (a) 1 (b) 2 (c) 4 (d) 3
  7. 7.What is the smallest number you must add to 113 to make it divisible by 7? (a) 5 (b) 4 (c) 6 (d) 1
  8. 8.Which of these numbers is divisible by 6? (a) 3862 (b) 3873 (c) 3864 (d) 3867
  9. 9.Which of these numbers is divisible by 4? (a) 2771 (b) 2775 (c) 2777 (d) 2772
  10. 10.What is the smallest number you must add to 805 to make it divisible by 11? (a) 9 (b) 10 (c) 2 (d) 8
  11. 11.What is the remainder when 111 is divided by 11? (a) 0 (b) 10 (c) 1 (d) 2
  12. 12.What is the smallest number you must add to 424 to make it divisible by 6? (a) 1 (b) 3 (c) 2 (d) 4
  13. 13.Which of these numbers is divisible by 9? (a) 2158 (b) 2168 (c) 2144 (d) 2151
  14. 14.Which of these numbers is divisible by 6? (a) 956 (b) 964 (c) 970 (d) 960
  15. 15.Which of these numbers is divisible by 4? (a) 3334 (b) 3328 (c) 3325 (d) 3329
  16. 16.What is the units digit of 4^43? (a) 3 (b) 4 (c) 5 (d) 6
  17. 17.What is the smallest number you must add to 714 to make it divisible by 11? (a) 10 (b) 3 (c) 2 (d) 1
  18. 18.What is the smallest number you must add to 292 to make it divisible by 11? (a) 7 (b) 6 (c) 5 (d) 4
  19. 19.What is the remainder when 119 is divided by 8? (a) 7 (b) 6 (c) 5 (d) 1
  20. 20.What is the units digit of 7^48? (a) 3 (b) 9 (c) 7 (d) 1
  21. 21.What is the smallest number you must add to 358 to make it divisible by 8? (a) 6 (b) 3 (c) 1 (d) 2
  22. 22.Which of these numbers is divisible by 3? (a) 650 (b) 655 (c) 651 (d) 652
  23. 23.Which of these numbers is divisible by 3? (a) 851 (b) 848 (c) 849 (d) 853
  24. 24.Which of these numbers is divisible by 3? (a) 1455 (b) 1453 (c) 1459 (d) 1456
  25. 25.Which of these numbers is divisible by 6? (a) 1290 (b) 1299 (c) 1295 (d) 1285
  26. 26.Which of these numbers is divisible by 8? (a) 5931 (b) 5920 (c) 5919 (d) 5923
  27. 27.What is the smallest number you must add to 408 to make it divisible by 10? (a) 3 (b) 1 (c) 2 (d) 8
  28. 28.What is the remainder when 46 is divided by 3? (a) 1 (b) 2 (c) 0
  29. 29.What is the remainder when 156 is divided by 8? (a) 4 (b) 5 (c) 3 (d) 6
  30. 30.What is the smallest number you must add to 55 to make it divisible by 4? (a) 1 (b) 2 (c) 3
  31. 31.Which of these numbers is divisible by 6? (a) 2387 (b) 2376 (c) 2372 (d) 2378
  32. 32.What is the units digit of 3^44? (a) 3 (b) 9 (c) 1 (d) 7
  33. 33.What is the smallest number you must add to 357 to make it divisible by 8? (a) 2 (b) 4 (c) 5 (d) 3
  34. 34.Which of these numbers is divisible by 8? (a) 2214 (b) 2216 (c) 2218 (d) 2228
  35. 35.Which of these numbers is divisible by 8? (a) 4889 (b) 4875 (c) 4882 (d) 4880
  36. 36.What is the units digit of 7^58? (a) 1 (b) 9 (c) 3 (d) 7
  37. 37.What is the remainder when 226 is divided by 4? (a) 0 (b) 1 (c) 2 (d) 3
  38. 38.What is the smallest number you must add to 110 to make it divisible by 6? (a) 5 (b) 3 (c) 4 (d) 2
  39. 39.Which of these numbers is divisible by 4? (a) 1178 (b) 1172 (c) 1174 (d) 1170
  40. 40.What is the smallest number you must add to 124 to make it divisible by 3? (a) 2 (b) 1

Answer key

  1. 9
  2. 141
  3. 1
  4. 1816
  5. 3
  6. 4
  7. 6
  8. 3864
  9. 2772
  10. 9
  11. 1
  12. 2
  13. 2151
  14. 960
  15. 3328
  16. 4
  17. 1
  18. 5
  19. 7
  20. 1
  21. 2
  22. 651
  23. 849
  24. 1455
  25. 1290
  26. 5920
  27. 2
  28. 1
  29. 4
  30. 1
  31. 2376
  32. 1
  33. 3
  34. 2216
  35. 4880
  36. 9
  37. 2
  38. 4
  39. 1172
  40. 2

Free maths olympiad lessons, practice and worksheets at talentjr.in/maths-olympiad