Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 704 · 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 4? (a) 1905 (b) 1904 (c) 1903 (d) 1911
2. 2.What is the remainder when 407 is divided by 7? (a) 6 (b) 0 (c) 2 (d) 1
3. 3.What is the units digit of 7^13? (a) 7 (b) 3 (c) 1 (d) 9
4. 4.What is the smallest number you must add to 591 to make it divisible by 8? (a) 2 (b) 3 (c) 1 (d) 7
5. 5.Which of these numbers is divisible by 8? (a) 2569 (b) 2557 (c) 2562 (d) 2560
6. 6.Which of these numbers is divisible by 4? (a) 2587 (b) 2590 (c) 2582 (d) 2584
7. 7.What is the smallest number you must add to 277 to make it divisible by 6? (a) 3 (b) 5 (c) 1 (d) 4
8. 8.Which of these numbers is divisible by 3? (a) 1386 (b) 1385 (c) 1391 (d) 1388
9. 9.What is the smallest number you must add to 750 to make it divisible by 11? (a) 2 (b) 10 (c) 8 (d) 9
10. 10.What is the smallest number you must add to 199 to make it divisible by 10? (a) 9 (b) 3 (c) 2 (d) 1
11. 11.Which of these numbers is divisible by 8? (a) 405 (b) 391 (c) 392 (d) 393
12. 12.Which of these numbers is divisible by 3? (a) 467 (b) 460 (c) 463 (d) 462
13. 13.What is the smallest number you must add to 812 to make it divisible by 11? (a) 2 (b) 3 (c) 1 (d) 9
14. 14.What is the remainder when 249 is divided by 4? (a) 2 (b) 3 (c) 0 (d) 1
15. 15.What is the remainder when 483 is divided by 6? (a) 4 (b) 2 (c) 5 (d) 3
16. 16.What is the units digit of 2^20? (a) 4 (b) 2 (c) 6 (d) 8
17. 17.What is the remainder when 367 is divided by 10? (a) 8 (b) 6 (c) 7 (d) 3
18. 18.What is the remainder when 54 is divided by 4? (a) 0 (b) 2 (c) 3 (d) 1
19. 19.What is the units digit of 9^45? (a) 1 (b) 9 (c) 7 (d) 8
20. 20.What is the units digit of 3^59? (a) 9 (b) 1 (c) 7 (d) 3
21. 21.What is the remainder when 53 is divided by 4? (a) 3 (b) 0 (c) 2 (d) 1
22. 22.What is the remainder when 318 is divided by 4? (a) 1 (b) 0 (c) 3 (d) 2
23. 23.What is the remainder when 65 is divided by 3? (a) 2 (b) 1 (c) 0
24. 24.Which of these numbers is divisible by 4? (a) 677 (b) 671 (c) 672 (d) 674
25. 25.What is the smallest number you must add to 629 to make it divisible by 8? (a) 5 (b) 3 (c) 2 (d) 4
26. 26.What is the units digit of 2^32? (a) 2 (b) 6 (c) 4 (d) 8
27. 27.What is the remainder when 547 is divided by 7? (a) 0 (b) 2 (c) 1 (d) 6
28. 28.What is the units digit of 9^27? (a) 1 (b) 8 (c) 7 (d) 9
29. 29.What is the remainder when 66 is divided by 4? (a) 3 (b) 2 (c) 1 (d) 0
30. 30.What is the smallest number you must add to 365 to make it divisible by 6? (a) 1 (b) 5 (c) 3 (d) 2
31. 31.What is the smallest number you must add to 66 to make it divisible by 4? (a) 2 (b) 3 (c) 1
32. 32.What is the units digit of 3^13? (a) 7 (b) 1 (c) 9 (d) 3
33. 33.What is the units digit of 4^17? (a) 3 (b) 4 (c) 6 (d) 5
34. 34.What is the units digit of 9^28? (a) 1 (b) 0 (c) 9 (d) 2
35. 35.What is the smallest number you must add to 182 to make it divisible by 3? (a) 2 (b) 1
36. 36.Which of these numbers is divisible by 6? (a) 3232 (b) 3234 (c) 3237 (d) 3245
37. 37.What is the units digit of 8^54? (a) 6 (b) 2 (c) 8 (d) 4
38. 38.What is the smallest number you must add to 362 to make it divisible by 10? (a) 9 (b) 8 (c) 7 (d) 2
39. 39.What is the units digit of 9^13? (a) 1 (b) 8 (c) 9 (d) 7
40. 40.Which of these numbers is divisible by 6? (a) 1046 (b) 1043 (c) 1051 (d) 1044

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