Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 928 · 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 244 is divided by 11? (a) 9 (b) 1 (c) 2 (d) 3
2. 2.Which of these numbers is divisible by 9? (a) 5646 (b) 5655 (c) 5652 (d) 5664
3. 3.What is the remainder when 115 is divided by 3? (a) 2 (b) 0 (c) 1
4. 4.Which of these numbers is divisible by 3? (a) 964 (b) 966 (c) 967 (d) 971
5. 5.Which of these numbers is divisible by 9? (a) 5202 (b) 5209 (c) 5213 (d) 5200
6. 6.Which of these numbers is divisible by 8? (a) 4368 (b) 4366 (c) 4382 (d) 4374
7. 7.What is the smallest number you must add to 364 to make it divisible by 11? (a) 8 (b) 10 (c) 1 (d) 9
8. 8.Which of these numbers is divisible by 9? (a) 3737 (b) 3733 (c) 3721 (d) 3726
9. 9.What is the smallest number you must add to 79 to make it divisible by 7? (a) 4 (b) 6 (c) 5 (d) 2
10. 10.What is the smallest number you must add to 192 to make it divisible by 5? (a) 4 (b) 2 (c) 1 (d) 3
11. 11.What is the smallest number you must add to 173 to make it divisible by 3? (a) 1 (b) 2
12. 12.What is the smallest number you must add to 358 to make it divisible by 10? (a) 8 (b) 2 (c) 1 (d) 3
13. 13.Which of these numbers is divisible by 8? (a) 746 (b) 759 (c) 752 (d) 764
14. 14.What is the remainder when 891 is divided by 12? (a) 2 (b) 9 (c) 3 (d) 4
15. 15.Which of these numbers is divisible by 4? (a) 351 (b) 353 (c) 352 (d) 358
16. 16.What is the units digit of 3^57? (a) 7 (b) 3 (c) 1 (d) 9
17. 17.What is the units digit of 7^42? (a) 1 (b) 9 (c) 7 (d) 3
18. 18.Which of these numbers is divisible by 9? (a) 2466 (b) 2474 (c) 2458 (d) 2476
19. 19.What is the remainder when 133 is divided by 11? (a) 0 (b) 1 (c) 10 (d) 2
20. 20.Which of these numbers is divisible by 6? (a) 2623 (b) 2616 (c) 2617 (d) 2611
21. 21.What is the units digit of 9^29? (a) 9 (b) 7 (c) 1 (d) 8
22. 22.Which of these numbers is divisible by 11? (a) 7533 (b) 7548 (c) 7535 (d) 7542
23. 23.Which of these numbers is divisible by 9? (a) 7184 (b) 7193 (c) 7205 (d) 7191
24. 24.What is the remainder when 345 is divided by 6? (a) 5 (b) 3 (c) 4 (d) 2
25. 25.What is the units digit of 8^43? (a) 2 (b) 6 (c) 4 (d) 8
26. 26.What is the units digit of 2^35? (a) 2 (b) 6 (c) 4 (d) 8
27. 27.What is the smallest number you must add to 806 to make it divisible by 11? (a) 9 (b) 7 (c) 8 (d) 3
28. 28.What is the remainder when 389 is divided by 7? (a) 3 (b) 6 (c) 5 (d) 4
29. 29.Which of these numbers is divisible by 8? (a) 1416 (b) 1426 (c) 1423 (d) 1414
30. 30.What is the remainder when 550 is divided by 7? (a) 3 (b) 5 (c) 6 (d) 4
31. 31.Which of these numbers is divisible by 11? (a) 4251 (b) 4269 (c) 4264 (d) 4257
32. 32.What is the smallest number you must add to 127 to make it divisible by 3? (a) 2 (b) 1
33. 33.Which of these numbers is divisible by 4? (a) 889 (b) 892 (c) 893 (d) 898
34. 34.What is the units digit of 9^36? (a) 9 (b) 2 (c) 0 (d) 1
35. 35.Which of these numbers is divisible by 3? (a) 308 (b) 311 (c) 309 (d) 314
36. 36.Which of these numbers is divisible by 9? (a) 1078 (b) 1062 (c) 1059 (d) 1067
37. 37.What is the remainder when 322 is divided by 8? (a) 6 (b) 3 (c) 1 (d) 2
38. 38.Which of these numbers is divisible by 8? (a) 6288 (b) 6291 (c) 6302 (d) 6285
39. 39.Which of these numbers is divisible by 9? (a) 5060 (b) 5058 (c) 5056 (d) 5071
40. 40.What is the smallest number you must add to 71 to make it divisible by 4? (a) 2 (b) 1 (c) 3

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