Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 895 · 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 9^14? (a) 0 (b) 1 (c) 2 (d) 9
2. 2.What is the smallest number you must add to 115 to make it divisible by 4? (a) 1 (b) 3 (c) 2
3. 3.What is the smallest number you must add to 597 to make it divisible by 12? (a) 9 (b) 3 (c) 4 (d) 2
4. 4.Which of these numbers is divisible by 4? (a) 2066 (b) 2057 (c) 2063 (d) 2060
5. 5.What is the smallest number you must add to 312 to make it divisible by 11? (a) 4 (b) 7 (c) 8 (d) 6
6. 6.What is the smallest number you must add to 267 to make it divisible by 10? (a) 4 (b) 2 (c) 3 (d) 7
7. 7.Which of these numbers is divisible by 11? (a) 5665 (b) 5660 (c) 5667 (d) 5686
8. 8.What is the units digit of 3^59? (a) 3 (b) 7 (c) 9 (d) 1
9. 9.Which of these numbers is divisible by 4? (a) 1383 (b) 1384 (c) 1387 (d) 1389
10. 10.What is the units digit of 8^53? (a) 8 (b) 4 (c) 2 (d) 6
11. 11.What is the smallest number you must add to 232 to make it divisible by 11? (a) 8 (b) 1 (c) 10 (d) 9
12. 12.Which of these numbers is divisible by 4? (a) 757 (b) 754 (c) 756 (d) 762
13. 13.What is the smallest number you must add to 52 to make it divisible by 3? (a) 1 (b) 2
14. 14.Which of these numbers is divisible by 4? (a) 1021 (b) 1016 (c) 1018 (d) 1013
15. 15.What is the units digit of 8^13? (a) 8 (b) 6 (c) 2 (d) 4
16. 16.What is the remainder when 164 is divided by 6? (a) 1 (b) 4 (c) 3 (d) 2
17. 17.What is the units digit of 8^24? (a) 6 (b) 2 (c) 4 (d) 8
18. 18.Which of these numbers is divisible by 6? (a) 4166 (b) 4163 (c) 4164 (d) 4172
19. 19.What is the remainder when 158 is divided by 4? (a) 3 (b) 2 (c) 0 (d) 1
20. 20.Which of these numbers is divisible by 8? (a) 3956 (b) 3941 (c) 3944 (d) 3951
21. 21.Which of these numbers is divisible by 3? (a) 493 (b) 490 (c) 497 (d) 492
22. 22.What is the units digit of 9^53? (a) 8 (b) 1 (c) 9 (d) 7
23. 23.Which of these numbers is divisible by 11? (a) 4498 (b) 4480 (c) 4500 (d) 4488
24. 24.What is the smallest number you must add to 542 to make it divisible by 11? (a) 3 (b) 7 (c) 9 (d) 8
25. 25.Which of these numbers is divisible by 9? (a) 3710 (b) 3725 (c) 3704 (d) 3708
26. 26.What is the smallest number you must add to 709 to make it divisible by 9? (a) 1 (b) 3 (c) 2 (d) 7
27. 27.What is the units digit of 8^33? (a) 4 (b) 8 (c) 6 (d) 2
28. 28.What is the units digit of 9^41? (a) 1 (b) 7 (c) 8 (d) 9
29. 29.What is the units digit of 7^19? (a) 1 (b) 3 (c) 9 (d) 7
30. 30.What is the remainder when 109 is divided by 4? (a) 1 (b) 2 (c) 3 (d) 0
31. 31.What is the smallest number you must add to 485 to make it divisible by 8? (a) 4 (b) 5 (c) 3 (d) 2
32. 32.Which of these numbers is divisible by 4? (a) 2374 (b) 2372 (c) 2377 (d) 2369
33. 33.Which of these numbers is divisible by 9? (a) 1280 (b) 1276 (c) 1269 (d) 1268
34. 34.Which of these numbers is divisible by 6? (a) 525 (b) 516 (c) 513 (d) 519
35. 35.What is the smallest number you must add to 203 to make it divisible by 8? (a) 6 (b) 3 (c) 5 (d) 4
36. 36.What is the units digit of 4^44? (a) 7 (b) 6 (c) 4 (d) 5
37. 37.Which of these numbers is divisible by 4? (a) 501 (b) 495 (c) 497 (d) 496
38. 38.What is the smallest number you must add to 318 to make it divisible by 5? (a) 2 (b) 4 (c) 3 (d) 1
39. 39.What is the units digit of 8^11? (a) 4 (b) 6 (c) 8 (d) 2
40. 40.What is the remainder when 521 is divided by 8? (a) 2 (b) 0 (c) 7 (d) 1

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