Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 476 · 40 questions

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.

1. 1.What is the units digit of 9^27? (a) 9 (b) 8 (c) 7 (d) 1
2. 2.Which of these numbers is divisible by 8? (a) 5460 (b) 5473 (c) 5464 (d) 5471
3. 3.What is the smallest number you must add to 205 to make it divisible by 8? (a) 3 (b) 5 (c) 2 (d) 4
4. 4.What is the units digit of 2^23? (a) 2 (b) 6 (c) 4 (d) 8
5. 5.What is the smallest number you must add to 372 to make it divisible by 7? (a) 4 (b) 6 (c) 5 (d) 1
6. 6.What is the smallest number you must add to 45 to make it divisible by 4? (a) 1 (b) 2 (c) 3
7. 7.What is the remainder when 73 is divided by 4? (a) 2 (b) 3 (c) 0 (d) 1
8. 8.What is the smallest number you must add to 355 to make it divisible by 10? (a) 7 (b) 6 (c) 4 (d) 5
9. 9.Which of these numbers is divisible by 11? (a) 514 (b) 486 (c) 495 (d) 501
10. 10.What is the units digit of 3^58? (a) 9 (b) 7 (c) 3 (d) 1
11. 11.Which of these numbers is divisible by 4? (a) 411 (b) 415 (c) 408 (d) 405
12. 12.Which of these numbers is divisible by 3? (a) 1733 (b) 1729 (c) 1735 (d) 1731
13. 13.What is the smallest number you must add to 654 to make it divisible by 11? (a) 7 (b) 5 (c) 8 (d) 6
14. 14.What is the smallest number you must add to 796 to make it divisible by 11? (a) 6 (b) 8 (c) 7 (d) 4
15. 15.What is the remainder when 798 is divided by 11? (a) 8 (b) 5 (c) 7 (d) 6
16. 16.What is the smallest number you must add to 139 to make it divisible by 3? (a) 2 (b) 1
17. 17.Which of these numbers is divisible by 9? (a) 6108 (b) 6098 (c) 6087 (d) 6093
18. 18.Which of these numbers is divisible by 8? (a) 1265 (b) 1264 (c) 1260 (d) 1279
19. 19.What is the remainder when 415 is divided by 8? (a) 7 (b) 1 (c) 5 (d) 6
20. 20.Which of these numbers is divisible by 11? (a) 8491 (b) 8474 (c) 8460 (d) 8470
21. 21.What is the remainder when 857 is divided by 12? (a) 6 (b) 5 (c) 7 (d) 4
22. 22.Which of these numbers is divisible by 8? (a) 5040 (b) 5044 (c) 5035 (d) 5054
23. 23.Which of these numbers is divisible by 4? (a) 669 (b) 674 (c) 668 (d) 666
24. 24.What is the remainder when 295 is divided by 6? (a) 5 (b) 1 (c) 2 (d) 0
25. 25.What is the smallest number you must add to 793 to make it divisible by 10? (a) 7 (b) 3 (c) 6 (d) 8
26. 26.What is the units digit of 9^11? (a) 7 (b) 9 (c) 8 (d) 1
27. 27.What is the remainder when 485 is divided by 12? (a) 6 (b) 5 (c) 7 (d) 4
28. 28.What is the smallest number you must add to 338 to make it divisible by 6? (a) 2 (b) 5 (c) 4 (d) 3
29. 29.What is the units digit of 4^25? (a) 5 (b) 4 (c) 3 (d) 6
30. 30.What is the remainder when 239 is divided by 3? (a) 0 (b) 1 (c) 2
31. 31.Which of these numbers is divisible by 9? (a) 2731 (b) 2722 (c) 2718 (d) 2711
32. 32.Which of these numbers is divisible by 8? (a) 1813 (b) 1816 (c) 1823 (d) 1830
33. 33.What is the units digit of 8^13? (a) 4 (b) 8 (c) 6 (d) 2
34. 34.What is the smallest number you must add to 261 to make it divisible by 4? (a) 3 (b) 2 (c) 1
35. 35.What is the remainder when 504 is divided by 11? (a) 8 (b) 10 (c) 2 (d) 9
36. 36.What is the remainder when 348 is divided by 8? (a) 4 (b) 3 (c) 6 (d) 5
37. 37.What is the units digit of 2^35? (a) 2 (b) 8 (c) 6 (d) 4
38. 38.What is the units digit of 4^36? (a) 7 (b) 6 (c) 4 (d) 5
39. 39.Which of these numbers is divisible by 9? (a) 4292 (b) 4298 (c) 4308 (d) 4293
40. 40.What is the remainder when 196 is divided by 5? (a) 1 (b) 2 (c) 0 (d) 4

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