Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 114 · 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 85 is divided by 3? (a) 2 (b) 1 (c) 0
2. 2.Which of these numbers is divisible by 3? (a) 167 (b) 170 (c) 173 (d) 168
3. 3.What is the smallest number you must add to 359 to make it divisible by 6? (a) 2 (b) 1 (c) 5 (d) 3
4. 4.Which of these numbers is divisible by 8? (a) 5510 (b) 5523 (c) 5518 (d) 5512
5. 5.What is the units digit of 7^25? (a) 7 (b) 9 (c) 1 (d) 3
6. 6.What is the remainder when 647 is divided by 10? (a) 3 (b) 6 (c) 8 (d) 7
7. 7.What is the smallest number you must add to 480 to make it divisible by 9? (a) 3 (b) 5 (c) 6 (d) 7
8. 8.What is the smallest number you must add to 724 to make it divisible by 10? (a) 5 (b) 6 (c) 4 (d) 7
9. 9.What is the smallest number you must add to 309 to make it divisible by 9? (a) 7 (b) 5 (c) 3 (d) 6
10. 10.What is the remainder when 797 is divided by 11? (a) 7 (b) 5 (c) 6 (d) 4
11. 11.What is the remainder when 199 is divided by 3? (a) 0 (b) 2 (c) 1
12. 12.What is the units digit of 3^56? (a) 9 (b) 7 (c) 1 (d) 3
13. 13.What is the smallest number you must add to 476 to make it divisible by 8? (a) 3 (b) 6 (c) 5 (d) 4
14. 14.Which of these numbers is divisible by 4? (a) 2705 (b) 2715 (c) 2710 (d) 2708
15. 15.Which of these numbers is divisible by 9? (a) 8029 (b) 8022 (c) 8019 (d) 8016
16. 16.What is the remainder when 118 is divided by 8? (a) 6 (b) 5 (c) 7 (d) 2
17. 17.Which of these numbers is divisible by 3? (a) 2418 (b) 2420 (c) 2416 (d) 2422
18. 18.What is the remainder when 111 is divided by 9? (a) 6 (b) 2 (c) 4 (d) 3
19. 19.Which of these numbers is divisible by 9? (a) 2402 (b) 2383 (c) 2385 (d) 2392
20. 20.Which of these numbers is divisible by 6? (a) 1241 (b) 1244 (c) 1236 (d) 1232
21. 21.What is the smallest number you must add to 62 to make it divisible by 5? (a) 2 (b) 4 (c) 3 (d) 1
22. 22.What is the units digit of 4^31? (a) 5 (b) 4 (c) 6 (d) 3
23. 23.What is the remainder when 844 is divided by 12? (a) 4 (b) 5 (c) 8 (d) 3
24. 24.What is the remainder when 241 is divided by 4? (a) 3 (b) 0 (c) 2 (d) 1
25. 25.Which of these numbers is divisible by 6? (a) 3998 (b) 3988 (c) 3990 (d) 3991
26. 26.What is the remainder when 409 is divided by 8? (a) 7 (b) 0 (c) 2 (d) 1
27. 27.What is the remainder when 49 is divided by 4? (a) 2 (b) 1 (c) 3 (d) 0
28. 28.What is the smallest number you must add to 205 to make it divisible by 10? (a) 6 (b) 4 (c) 7 (d) 5
29. 29.What is the units digit of 7^17? (a) 1 (b) 9 (c) 3 (d) 7
30. 30.What is the units digit of 4^53? (a) 6 (b) 5 (c) 4 (d) 3
31. 31.Which of these numbers is divisible by 4? (a) 773 (b) 777 (c) 781 (d) 776
32. 32.What is the remainder when 364 is divided by 9? (a) 4 (b) 3 (c) 6 (d) 5
33. 33.What is the units digit of 8^60? (a) 2 (b) 6 (c) 4 (d) 8
34. 34.What is the remainder when 214 is divided by 3? (a) 0 (b) 2 (c) 1
35. 35.What is the remainder when 293 is divided by 4? (a) 3 (b) 1 (c) 0 (d) 2
36. 36.Which of these numbers is divisible by 6? (a) 502 (b) 496 (c) 498 (d) 509
37. 37.What is the smallest number you must add to 258 to make it divisible by 4? (a) 2 (b) 1 (c) 3
38. 38.What is the remainder when 81 is divided by 4? (a) 3 (b) 2 (c) 0 (d) 1
39. 39.What is the smallest number you must add to 573 to make it divisible by 12? (a) 4 (b) 9 (c) 2 (d) 3
40. 40.Which of these numbers is divisible by 9? (a) 1197 (b) 1214 (c) 1193 (d) 1199

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