Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 795 · 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 353 is divided by 11? (a) 1 (b) 2 (c) 10 (d) 0
2. 2.What is the smallest number you must add to 345 to make it divisible by 11? (a) 8 (b) 4 (c) 6 (d) 7
3. 3.What is the remainder when 324 is divided by 5? (a) 4 (b) 1 (c) 3 (d) 2
4. 4.Which of these numbers is divisible by 4? (a) 2108 (b) 2107 (c) 2113 (d) 2111
5. 5.What is the units digit of 9^15? (a) 1 (b) 7 (c) 9 (d) 8
6. 6.What is the units digit of 4^18? (a) 7 (b) 6 (c) 5 (d) 4
7. 7.Which of these numbers is divisible by 8? (a) 2793 (b) 2811 (c) 2800 (d) 2802
8. 8.What is the units digit of 9^33? (a) 1 (b) 9 (c) 7 (d) 8
9. 9.Which of these numbers is divisible by 9? (a) 648 (b) 660 (c) 651 (d) 645
10. 10.What is the remainder when 71 is divided by 5? (a) 4 (b) 0 (c) 1 (d) 2
11. 11.What is the smallest number you must add to 566 to make it divisible by 9? (a) 3 (b) 1 (c) 8 (d) 2
12. 12.Which of these numbers is divisible by 9? (a) 6876 (b) 6873 (c) 6886 (d) 6881
13. 13.What is the remainder when 154 is divided by 9? (a) 0 (b) 8 (c) 1 (d) 2
14. 14.What is the units digit of 2^14? (a) 2 (b) 6 (c) 4 (d) 8
15. 15.What is the remainder when 445 is divided by 9? (a) 5 (b) 6 (c) 4 (d) 3
16. 16.What is the smallest number you must add to 449 to make it divisible by 9? (a) 3 (b) 2 (c) 1 (d) 8
17. 17.What is the remainder when 119 is divided by 3? (a) 1 (b) 2 (c) 0
18. 18.What is the units digit of 8^51? (a) 4 (b) 2 (c) 8 (d) 6
19. 19.What is the smallest number you must add to 253 to make it divisible by 9? (a) 1 (b) 7 (c) 6 (d) 8
20. 20.What is the units digit of 4^10? (a) 7 (b) 6 (c) 5 (d) 4
21. 21.What is the remainder when 74 is divided by 6? (a) 1 (b) 3 (c) 2 (d) 4
22. 22.What is the remainder when 170 is divided by 11? (a) 7 (b) 4 (c) 6 (d) 5
23. 23.What is the remainder when 302 is divided by 10? (a) 3 (b) 8 (c) 1 (d) 2
24. 24.What is the smallest number you must add to 449 to make it divisible by 8? (a) 6 (b) 1 (c) 7 (d) 5
25. 25.What is the units digit of 4^57? (a) 6 (b) 5 (c) 3 (d) 4
26. 26.What is the remainder when 288 is divided by 5? (a) 1 (b) 2 (c) 4 (d) 3
27. 27.Which of these numbers is divisible by 6? (a) 525 (b) 528 (c) 533 (d) 536
28. 28.Which of these numbers is divisible by 6? (a) 3876 (b) 3880 (c) 3884 (d) 3873
29. 29.Which of these numbers is divisible by 4? (a) 3596 (b) 3595 (c) 3602 (d) 3599
30. 30.What is the remainder when 230 is divided by 6? (a) 3 (b) 4 (c) 2 (d) 1
31. 31.What is the remainder when 399 is divided by 8? (a) 7 (b) 5 (c) 6 (d) 1
32. 32.What is the remainder when 258 is divided by 5? (a) 2 (b) 4 (c) 1 (d) 3
33. 33.Which of these numbers is divisible by 3? (a) 777 (b) 775 (c) 778 (d) 782
34. 34.What is the remainder when 161 is divided by 5? (a) 1 (b) 0 (c) 2 (d) 4
35. 35.Which of these numbers is divisible by 3? (a) 719 (b) 725 (c) 721 (d) 720
36. 36.What is the smallest number you must add to 508 to make it divisible by 11? (a) 2 (b) 10 (c) 8 (d) 9
37. 37.What is the smallest number you must add to 329 to make it divisible by 5? (a) 4 (b) 2 (c) 1 (d) 3
38. 38.What is the smallest number you must add to 372 to make it divisible by 5? (a) 2 (b) 4 (c) 1 (d) 3
39. 39.Which of these numbers is divisible by 4? (a) 2282 (b) 2286 (c) 2284 (d) 2291
40. 40.What is the remainder when 291 is divided by 7? (a) 5 (b) 4 (c) 3 (d) 6

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