Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 40 · 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 remainder when 344 is divided by 11? (a) 8 (b) 3 (c) 2 (d) 4
2. 2.What is the remainder when 311 is divided by 7? (a) 3 (b) 5 (c) 4 (d) 2
3. 3.What is the units digit of 8^27? (a) 2 (b) 4 (c) 8 (d) 6
4. 4.What is the remainder when 223 is divided by 5? (a) 4 (b) 1 (c) 2 (d) 3
5. 5.Which of these numbers is divisible by 3? (a) 1238 (b) 1240 (c) 1244 (d) 1239
6. 6.What is the smallest number you must add to 566 to make it divisible by 8? (a) 1 (b) 3 (c) 6 (d) 2
7. 7.What is the units digit of 8^30? (a) 8 (b) 6 (c) 2 (d) 4
8. 8.Which of these numbers is divisible by 11? (a) 2711 (b) 2727 (c) 2717 (d) 2738
9. 9.What is the units digit of 2^11? (a) 6 (b) 4 (c) 8 (d) 2
10. 10.What is the smallest number you must add to 637 to make it divisible by 8? (a) 4 (b) 3 (c) 5 (d) 2
11. 11.Which of these numbers is divisible by 8? (a) 5273 (b) 5292 (c) 5280 (d) 5282
12. 12.Which of these numbers is divisible by 3? (a) 429 (b) 427 (c) 431 (d) 434
13. 13.Which of these numbers is divisible by 11? (a) 6989 (b) 6972 (c) 6974 (d) 6984
14. 14.Which of these numbers is divisible by 6? (a) 4609 (b) 4619 (c) 4605 (d) 4608
15. 15.What is the remainder when 98 is divided by 4? (a) 1 (b) 0 (c) 2 (d) 3
16. 16.Which of these numbers is divisible by 4? (a) 2399 (b) 2392 (c) 2393 (d) 2389
17. 17.Which of these numbers is divisible by 11? (a) 6759 (b) 6772 (c) 6781 (d) 6765
18. 18.What is the remainder when 314 is divided by 12? (a) 2 (b) 10 (c) 1 (d) 3
19. 19.What is the units digit of 9^45? (a) 1 (b) 8 (c) 7 (d) 9
20. 20.Which of these numbers is divisible by 4? (a) 517 (b) 512 (c) 509 (d) 513
21. 21.What is the units digit of 8^52? (a) 6 (b) 2 (c) 4 (d) 8
22. 22.What is the smallest number you must add to 341 to make it divisible by 5? (a) 2 (b) 3 (c) 1 (d) 4
23. 23.What is the smallest number you must add to 195 to make it divisible by 10? (a) 4 (b) 6 (c) 5 (d) 7
24. 24.What is the remainder when 659 is divided by 9? (a) 3 (b) 2 (c) 7 (d) 1
25. 25.What is the units digit of 9^56? (a) 2 (b) 0 (c) 9 (d) 1
26. 26.What is the smallest number you must add to 186 to make it divisible by 11? (a) 3 (b) 2 (c) 10 (d) 1
27. 27.What is the units digit of 9^36? (a) 2 (b) 9 (c) 0 (d) 1
28. 28.Which of these numbers is divisible by 8? (a) 5544 (b) 5546 (c) 5558 (d) 5539
29. 29.What is the smallest number you must add to 149 to make it divisible by 3? (a) 1 (b) 2
30. 30.What is the smallest number you must add to 327 to make it divisible by 7? (a) 3 (b) 1 (c) 2 (d) 5
31. 31.What is the units digit of 3^12? (a) 9 (b) 3 (c) 7 (d) 1
32. 32.What is the units digit of 8^31? (a) 6 (b) 8 (c) 4 (d) 2
33. 33.What is the smallest number you must add to 236 to make it divisible by 8? (a) 6 (b) 5 (c) 4 (d) 3
34. 34.What is the remainder when 249 is divided by 4? (a) 1 (b) 0 (c) 3 (d) 2
35. 35.What is the remainder when 239 is divided by 6? (a) 1 (b) 4 (c) 3 (d) 5
36. 36.What is the units digit of 8^15? (a) 4 (b) 6 (c) 8 (d) 2
37. 37.What is the units digit of 4^33? (a) 6 (b) 4 (c) 3 (d) 5
38. 38.What is the remainder when 452 is divided by 12? (a) 8 (b) 9 (c) 7 (d) 4
39. 39.What is the units digit of 7^24? (a) 7 (b) 9 (c) 3 (d) 1
40. 40.What is the units digit of 9^42? (a) 2 (b) 9 (c) 0 (d) 1

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