Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 998 · 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 8^17? (a) 6 (b) 8 (c) 2 (d) 4
2. 2.What is the remainder when 589 is divided by 8? (a) 4 (b) 3 (c) 6 (d) 5
3. 3.What is the remainder when 687 is divided by 10? (a) 8 (b) 3 (c) 6 (d) 7
4. 4.What is the remainder when 786 is divided by 12? (a) 6 (b) 8 (c) 5 (d) 7
5. 5.What is the remainder when 94 is divided by 5? (a) 4 (b) 2 (c) 3 (d) 1
6. 6.Which of these numbers is divisible by 3? (a) 1337 (b) 1340 (c) 1342 (d) 1338
7. 7.Which of these numbers is divisible by 9? (a) 1571 (b) 1554 (c) 1557 (d) 1563
8. 8.What is the remainder when 231 is divided by 6? (a) 4 (b) 2 (c) 3 (d) 5
9. 9.What is the smallest number you must add to 573 to make it divisible by 11? (a) 1 (b) 10 (c) 8 (d) 9
10. 10.What is the remainder when 233 is divided by 12? (a) 5 (b) 4 (c) 7 (d) 6
11. 11.Which of these numbers is divisible by 8? (a) 5371 (b) 5363 (c) 5360 (d) 5354
12. 12.Which of these numbers is divisible by 8? (a) 6510 (b) 6515 (c) 6501 (d) 6504
13. 13.What is the remainder when 613 is divided by 9? (a) 0 (b) 2 (c) 8 (d) 1
14. 14.What is the units digit of 4^10? (a) 6 (b) 5 (c) 4 (d) 7
15. 15.What is the remainder when 434 is divided by 6? (a) 4 (b) 2 (c) 1 (d) 3
16. 16.Which of these numbers is divisible by 3? (a) 817 (b) 820 (c) 816 (d) 815
17. 17.Which of these numbers is divisible by 9? (a) 502 (b) 495 (c) 492 (d) 506
18. 18.Which of these numbers is divisible by 8? (a) 2578 (b) 2590 (c) 2570 (d) 2576
19. 19.What is the units digit of 2^10? (a) 4 (b) 8 (c) 2 (d) 6
20. 20.What is the units digit of 7^35? (a) 7 (b) 3 (c) 1 (d) 9
21. 21.What is the smallest number you must add to 221 to make it divisible by 12? (a) 7 (b) 8 (c) 6 (d) 5
22. 22.What is the remainder when 125 is divided by 4? (a) 3 (b) 2 (c) 1 (d) 0
23. 23.What is the remainder when 148 is divided by 5? (a) 1 (b) 2 (c) 3 (d) 4
24. 24.What is the remainder when 738 is divided by 12? (a) 8 (b) 6 (c) 5 (d) 7
25. 25.What is the smallest number you must add to 682 to make it divisible by 10? (a) 7 (b) 9 (c) 2 (d) 8
26. 26.What is the units digit of 4^15? (a) 5 (b) 4 (c) 6 (d) 3
27. 27.What is the smallest number you must add to 296 to make it divisible by 7? (a) 6 (b) 4 (c) 5 (d) 2
28. 28.Which of these numbers is divisible by 6? (a) 858 (b) 860 (c) 857 (d) 868
29. 29.Which of these numbers is divisible by 3? (a) 805 (b) 802 (c) 801 (d) 800
30. 30.What is the smallest number you must add to 102 to make it divisible by 4? (a) 3 (b) 1 (c) 2
31. 31.What is the smallest number you must add to 50 to make it divisible by 3? (a) 2 (b) 1
32. 32.What is the smallest number you must add to 127 to make it divisible by 3? (a) 2 (b) 1
33. 33.What is the remainder when 265 is divided by 10? (a) 6 (b) 4 (c) 5 (d) 7
34. 34.What is the units digit of 3^15? (a) 9 (b) 3 (c) 1 (d) 7
35. 35.Which of these numbers is divisible by 3? (a) 2681 (b) 2678 (c) 2679 (d) 2683
36. 36.Which of these numbers is divisible by 3? (a) 941 (b) 938 (c) 943 (d) 939
37. 37.What is the remainder when 353 is divided by 9? (a) 3 (b) 1 (c) 7 (d) 2
38. 38.What is the remainder when 417 is divided by 10? (a) 7 (b) 8 (c) 6 (d) 3
39. 39.Which of these numbers is divisible by 3? (a) 272 (b) 269 (c) 265 (d) 267
40. 40.What is the units digit of 7^36? (a) 7 (b) 1 (c) 3 (d) 9

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