Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 445 · 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 614 is divided by 12? (a) 2 (b) 3 (c) 1 (d) 10
2. 2.What is the smallest number you must add to 321 to make it divisible by 5? (a) 3 (b) 1 (c) 4 (d) 2
3. 3.Which of these numbers is divisible by 3? (a) 1054 (b) 1053 (c) 1051 (d) 1057
4. 4.What is the remainder when 362 is divided by 9? (a) 1 (b) 7 (c) 3 (d) 2
5. 5.What is the remainder when 583 is divided by 8? (a) 1 (b) 6 (c) 5 (d) 7
6. 6.What is the units digit of 2^30? (a) 8 (b) 4 (c) 6 (d) 2
7. 7.What is the units digit of 2^41? (a) 8 (b) 4 (c) 6 (d) 2
8. 8.Which of these numbers is divisible by 3? (a) 392 (b) 395 (c) 398 (d) 393
9. 9.What is the smallest number you must add to 499 to make it divisible by 7? (a) 6 (b) 4 (c) 5 (d) 2
10. 10.What is the remainder when 31 is divided by 3? (a) 0 (b) 2 (c) 1
11. 11.Which of these numbers is divisible by 6? (a) 2759 (b) 2763 (c) 2769 (d) 2760
12. 12.What is the units digit of 2^11? (a) 6 (b) 4 (c) 8 (d) 2
13. 13.What is the remainder when 192 is divided by 10? (a) 3 (b) 8 (c) 2 (d) 1
14. 14.What is the units digit of 8^23? (a) 4 (b) 2 (c) 6 (d) 8
15. 15.Which of these numbers is divisible by 3? (a) 2587 (b) 2584 (c) 2582 (d) 2583
16. 16.What is the remainder when 723 is divided by 10? (a) 4 (b) 7 (c) 3 (d) 2
17. 17.What is the units digit of 4^20? (a) 4 (b) 6 (c) 5 (d) 7
18. 18.Which of these numbers is divisible by 9? (a) 2396 (b) 2404 (c) 2403 (d) 2414
19. 19.What is the remainder when 610 is divided by 8? (a) 6 (b) 3 (c) 1 (d) 2
20. 20.What is the smallest number you must add to 417 to make it divisible by 12? (a) 2 (b) 9 (c) 3 (d) 4
21. 21.What is the remainder when 194 is divided by 7? (a) 2 (b) 6 (c) 4 (d) 5
22. 22.Which of these numbers is divisible by 6? (a) 1256 (b) 1249 (c) 1261 (d) 1254
23. 23.What is the remainder when 327 is divided by 8? (a) 1 (b) 7 (c) 6 (d) 5
24. 24.What is the units digit of 4^10? (a) 7 (b) 5 (c) 4 (d) 6
25. 25.What is the units digit of 2^13? (a) 2 (b) 8 (c) 4 (d) 6
26. 26.What is the units digit of 8^10? (a) 8 (b) 2 (c) 6 (d) 4
27. 27.Which of these numbers is divisible by 9? (a) 3868 (b) 3853 (c) 3847 (d) 3852
28. 28.What is the units digit of 4^54? (a) 5 (b) 7 (c) 6 (d) 4
29. 29.What is the remainder when 133 is divided by 3? (a) 0 (b) 1 (c) 2
30. 30.What is the units digit of 3^20? (a) 9 (b) 3 (c) 1 (d) 7
31. 31.What is the units digit of 9^40? (a) 1 (b) 0 (c) 9 (d) 2
32. 32.What is the units digit of 2^32? (a) 4 (b) 8 (c) 6 (d) 2
33. 33.What is the smallest number you must add to 262 to make it divisible by 5? (a) 2 (b) 1 (c) 3 (d) 4
34. 34.Which of these numbers is divisible by 9? (a) 975 (b) 969 (c) 959 (d) 963
35. 35.What is the remainder when 253 is divided by 5? (a) 2 (b) 3 (c) 4 (d) 1
36. 36.What is the remainder when 236 is divided by 3? (a) 0 (b) 2 (c) 1
37. 37.Which of these numbers is divisible by 3? (a) 499 (b) 496 (c) 502 (d) 498
38. 38.What is the remainder when 754 is divided by 10? (a) 3 (b) 5 (c) 4 (d) 6
39. 39.What is the smallest number you must add to 275 to make it divisible by 9? (a) 3 (b) 5 (c) 4 (d) 6
40. 40.Which of these numbers is divisible by 8? (a) 5530 (b) 5520 (c) 5518 (d) 5523

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