Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 193 · 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.Which of these numbers is divisible by 8? (a) 5914 (b) 5920 (c) 5924 (d) 5930
2. 2.Which of these numbers is divisible by 11? (a) 7230 (b) 7221 (c) 7227 (d) 7244
3. 3.What is the units digit of 2^28? (a) 2 (b) 8 (c) 6 (d) 4
4. 4.Which of these numbers is divisible by 4? (a) 2861 (b) 2859 (c) 2856 (d) 2854
5. 5.What is the smallest number you must add to 149 to make it divisible by 5? (a) 4 (b) 2 (c) 1 (d) 3
6. 6.What is the smallest number you must add to 731 to make it divisible by 11? (a) 7 (b) 5 (c) 6 (d) 8
7. 7.Which of these numbers is divisible by 3? (a) 1272 (b) 1276 (c) 1274 (d) 1271
8. 8.What is the remainder when 243 is divided by 7? (a) 6 (b) 2 (c) 4 (d) 5
9. 9.What is the units digit of 9^37? (a) 8 (b) 7 (c) 1 (d) 9
10. 10.What is the units digit of 4^58? (a) 4 (b) 5 (c) 6 (d) 7
11. 11.Which of these numbers is divisible by 6? (a) 496 (b) 485 (c) 486 (d) 487
12. 12.Which of these numbers is divisible by 11? (a) 3262 (b) 3245 (c) 3239 (d) 3246
13. 13.Which of these numbers is divisible by 9? (a) 4563 (b) 4580 (c) 4560 (d) 4569
14. 14.What is the smallest number you must add to 184 to make it divisible by 3? (a) 2 (b) 1
15. 15.Which of these numbers is divisible by 9? (a) 7488 (b) 7491 (c) 7503 (d) 7484
16. 16.What is the smallest number you must add to 175 to make it divisible by 3? (a) 1 (b) 2
17. 17.What is the smallest number you must add to 547 to make it divisible by 7? (a) 5 (b) 4 (c) 6 (d) 1
18. 18.Which of these numbers is divisible by 9? (a) 1773 (b) 1774 (c) 1766 (d) 1787
19. 19.What is the remainder when 306 is divided by 4? (a) 1 (b) 2 (c) 0 (d) 3
20. 20.Which of these numbers is divisible by 8? (a) 4512 (b) 4518 (c) 4522 (d) 4506
21. 21.What is the smallest number you must add to 161 to make it divisible by 3? (a) 1 (b) 2
22. 22.What is the remainder when 331 is divided by 5? (a) 4 (b) 0 (c) 1 (d) 2
23. 23.What is the units digit of 4^10? (a) 7 (b) 6 (c) 5 (d) 4
24. 24.What is the units digit of 8^56? (a) 4 (b) 2 (c) 8 (d) 6
25. 25.What is the units digit of 7^36? (a) 9 (b) 1 (c) 7 (d) 3
26. 26.What is the smallest number you must add to 374 to make it divisible by 10? (a) 6 (b) 4 (c) 7 (d) 5
27. 27.What is the remainder when 243 is divided by 4? (a) 1 (b) 0 (c) 3 (d) 2
28. 28.What is the remainder when 261 is divided by 4? (a) 3 (b) 2 (c) 0 (d) 1
29. 29.What is the smallest number you must add to 64 to make it divisible by 3? (a) 1 (b) 2
30. 30.What is the smallest number you must add to 442 to make it divisible by 7? (a) 5 (b) 4 (c) 1 (d) 6
31. 31.Which of these numbers is divisible by 11? (a) 2467 (b) 2453 (c) 2451 (d) 2456
32. 32.What is the remainder when 223 is divided by 3? (a) 0 (b) 1 (c) 2
33. 33.What is the remainder when 181 is divided by 5? (a) 2 (b) 1 (c) 4 (d) 0
34. 34.What is the smallest number you must add to 109 to make it divisible by 4? (a) 3 (b) 1 (c) 2
35. 35.What is the units digit of 2^58? (a) 8 (b) 4 (c) 2 (d) 6
36. 36.What is the smallest number you must add to 207 to make it divisible by 4? (a) 3 (b) 2 (c) 1
37. 37.What is the smallest number you must add to 101 to make it divisible by 6? (a) 3 (b) 2 (c) 1 (d) 5
38. 38.What is the smallest number you must add to 377 to make it divisible by 10? (a) 2 (b) 3 (c) 7 (d) 4
39. 39.Which of these numbers is divisible by 4? (a) 1406 (b) 1397 (c) 1400 (d) 1401
40. 40.What is the units digit of 3^52? (a) 9 (b) 1 (c) 3 (d) 7

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