← Divisibility, remainders and units digits lesson New set →

Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 436 · 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 128 to make it divisible by 6? (a) 4 (b) 3 (c) 5 (d) 2
  2. 2.What is the units digit of 2^45? (a) 6 (b) 8 (c) 2 (d) 4
  3. 3.What is the smallest number you must add to 233 to make it divisible by 5? (a) 1 (b) 3 (c) 4 (d) 2
  4. 4.Which of these numbers is divisible by 9? (a) 2543 (b) 2549 (c) 2530 (d) 2538
  5. 5.What is the remainder when 174 is divided by 5? (a) 1 (b) 3 (c) 2 (d) 4
  6. 6.What is the units digit of 9^13? (a) 9 (b) 8 (c) 7 (d) 1
  7. 7.What is the units digit of 4^51? (a) 4 (b) 3 (c) 6 (d) 5
  8. 8.What is the smallest number you must add to 391 to make it divisible by 5? (a) 3 (b) 1 (c) 4 (d) 2
  9. 9.Which of these numbers is divisible by 11? (a) 6891 (b) 6870 (c) 6875 (d) 6877
  10. 10.What is the units digit of 8^46? (a) 8 (b) 4 (c) 2 (d) 6
  11. 11.What is the remainder when 594 is divided by 10? (a) 3 (b) 5 (c) 6 (d) 4
  12. 12.What is the smallest number you must add to 299 to make it divisible by 5? (a) 3 (b) 1 (c) 2 (d) 4
  13. 13.What is the smallest number you must add to 369 to make it divisible by 6? (a) 5 (b) 4 (c) 2 (d) 3
  14. 14.Which of these numbers is divisible by 3? (a) 1440 (b) 1444 (c) 1442 (d) 1438
  15. 15.What is the smallest number you must add to 160 to make it divisible by 7? (a) 1 (b) 6 (c) 2 (d) 3
  16. 16.Which of these numbers is divisible by 4? (a) 1994 (b) 1987 (c) 1989 (d) 1988
  17. 17.What is the remainder when 549 is divided by 10? (a) 9 (b) 7 (c) 1 (d) 8
  18. 18.What is the units digit of 3^39? (a) 1 (b) 9 (c) 3 (d) 7
  19. 19.What is the remainder when 174 is divided by 4? (a) 1 (b) 3 (c) 0 (d) 2
  20. 20.What is the smallest number you must add to 423 to make it divisible by 6? (a) 4 (b) 2 (c) 5 (d) 3
  21. 21.Which of these numbers is divisible by 4? (a) 999 (b) 991 (c) 992 (d) 995
  22. 22.What is the smallest number you must add to 124 to make it divisible by 10? (a) 7 (b) 6 (c) 5 (d) 4
  23. 23.What is the smallest number you must add to 113 to make it divisible by 11? (a) 3 (b) 7 (c) 8 (d) 9
  24. 24.What is the smallest number you must add to 152 to make it divisible by 5? (a) 4 (b) 2 (c) 3 (d) 1
  25. 25.What is the remainder when 124 is divided by 6? (a) 2 (b) 4 (c) 3 (d) 5
  26. 26.Which of these numbers is divisible by 4? (a) 1477 (b) 1482 (c) 1487 (d) 1480
  27. 27.What is the units digit of 2^37? (a) 8 (b) 4 (c) 6 (d) 2
  28. 28.What is the remainder when 325 is divided by 12? (a) 0 (b) 11 (c) 2 (d) 1
  29. 29.What is the smallest number you must add to 883 to make it divisible by 12? (a) 4 (b) 5 (c) 6 (d) 7
  30. 30.Which of these numbers is divisible by 6? (a) 4995 (b) 5002 (c) 5006 (d) 4998
  31. 31.Which of these numbers is divisible by 6? (a) 326 (b) 324 (c) 320 (d) 333
  32. 32.What is the units digit of 3^38? (a) 7 (b) 9 (c) 3 (d) 1
  33. 33.What is the smallest number you must add to 89 to make it divisible by 3? (a) 2 (b) 1
  34. 34.Which of these numbers is divisible by 8? (a) 1529 (b) 1536 (c) 1548 (d) 1537
  35. 35.What is the smallest number you must add to 175 to make it divisible by 6? (a) 1 (b) 4 (c) 5 (d) 3
  36. 36.Which of these numbers is divisible by 8? (a) 2859 (b) 2856 (c) 2852 (d) 2869
  37. 37.Which of these numbers is divisible by 6? (a) 4388 (b) 4403 (c) 4392 (d) 4394
  38. 38.What is the remainder when 824 is divided by 12? (a) 4 (b) 9 (c) 7 (d) 8
  39. 39.Which of these numbers is divisible by 8? (a) 359 (b) 366 (c) 352 (d) 347
  40. 40.Which of these numbers is divisible by 4? (a) 2683 (b) 2675 (c) 2679 (d) 2676

Answer key

  1. 4
  2. 2
  3. 2
  4. 2538
  5. 4
  6. 9
  7. 4
  8. 4
  9. 6875
  10. 4
  11. 4
  12. 1
  13. 3
  14. 1440
  15. 1
  16. 1988
  17. 9
  18. 7
  19. 2
  20. 3
  21. 992
  22. 6
  23. 8
  24. 3
  25. 4
  26. 1480
  27. 2
  28. 1
  29. 5
  30. 4998
  31. 324
  32. 9
  33. 1
  34. 1536
  35. 5
  36. 2856
  37. 4392
  38. 8
  39. 352
  40. 2676

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