Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 441 · 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 46 is divided by 4? (a) 0 (b) 2 (c) 1 (d) 3
2. 2.What is the units digit of 7^55? (a) 3 (b) 9 (c) 1 (d) 7
3. 3.Which of these numbers is divisible by 3? (a) 400 (b) 394 (c) 397 (d) 396
4. 4.What is the smallest number you must add to 188 to make it divisible by 5? (a) 1 (b) 3 (c) 2 (d) 4
5. 5.Which of these numbers is divisible by 11? (a) 7978 (b) 7975 (c) 7972 (d) 7988
6. 6.What is the remainder when 181 is divided by 3? (a) 1 (b) 2 (c) 0
7. 7.Which of these numbers is divisible by 4? (a) 3418 (b) 3422 (c) 3416 (d) 3413
8. 8.What is the smallest number you must add to 738 to make it divisible by 10? (a) 1 (b) 8 (c) 3 (d) 2
9. 9.What is the remainder when 97 is divided by 4? (a) 0 (b) 3 (c) 1 (d) 2
10. 10.Which of these numbers is divisible by 8? (a) 3032 (b) 3039 (c) 3025 (d) 3046
11. 11.What is the remainder when 484 is divided by 10? (a) 3 (b) 4 (c) 6 (d) 5
12. 12.Which of these numbers is divisible by 4? (a) 3465 (b) 3464 (c) 3463 (d) 3469
13. 13.What is the remainder when 659 is divided by 11? (a) 1 (b) 10 (c) 9 (d) 8
14. 14.Which of these numbers is divisible by 9? (a) 5466 (b) 5484 (c) 5472 (d) 5473
15. 15.What is the units digit of 3^46? (a) 9 (b) 3 (c) 1 (d) 7
16. 16.What is the remainder when 721 is divided by 11? (a) 7 (b) 8 (c) 5 (d) 6
17. 17.What is the remainder when 747 is divided by 12? (a) 4 (b) 3 (c) 9 (d) 2
18. 18.What is the units digit of 4^11? (a) 3 (b) 5 (c) 6 (d) 4
19. 19.Which of these numbers is divisible by 4? (a) 753 (b) 759 (c) 756 (d) 763
20. 20.What is the remainder when 289 is divided by 4? (a) 0 (b) 2 (c) 1 (d) 3
21. 21.What is the units digit of 9^15? (a) 8 (b) 1 (c) 9 (d) 7
22. 22.Which of these numbers is divisible by 9? (a) 2739 (b) 2727 (c) 2719 (d) 2730
23. 23.Which of these numbers is divisible by 9? (a) 7785 (b) 7790 (c) 7798 (d) 7780
24. 24.What is the remainder when 169 is divided by 3? (a) 0 (b) 2 (c) 1
25. 25.What is the remainder when 160 is divided by 3? (a) 1 (b) 0 (c) 2
26. 26.What is the smallest number you must add to 454 to make it divisible by 7? (a) 2 (b) 6 (c) 1 (d) 3
27. 27.What is the units digit of 2^21? (a) 8 (b) 4 (c) 2 (d) 6
28. 28.What is the remainder when 637 is divided by 9? (a) 7 (b) 6 (c) 2 (d) 8
29. 29.What is the smallest number you must add to 209 to make it divisible by 3? (a) 1 (b) 2
30. 30.What is the smallest number you must add to 190 to make it divisible by 9? (a) 7 (b) 1 (c) 6 (d) 8
31. 31.What is the units digit of 8^55? (a) 6 (b) 4 (c) 8 (d) 2
32. 32.What is the units digit of 8^41? (a) 2 (b) 4 (c) 8 (d) 6
33. 33.What is the remainder when 123 is divided by 11? (a) 9 (b) 3 (c) 2 (d) 1
34. 34.Which of these numbers is divisible by 9? (a) 578 (b) 585 (c) 592 (d) 601
35. 35.What is the remainder when 279 is divided by 7? (a) 1 (b) 4 (c) 5 (d) 6
36. 36.What is the smallest number you must add to 321 to make it divisible by 4? (a) 2 (b) 3 (c) 1
37. 37.Which of these numbers is divisible by 6? (a) 492 (b) 500 (c) 497 (d) 490
38. 38.What is the remainder when 440 is divided by 6? (a) 2 (b) 4 (c) 1 (d) 3
39. 39.Which of these numbers is divisible by 8? (a) 5113 (b) 5097 (c) 5110 (d) 5104
40. 40.Which of these numbers is divisible by 4? (a) 1092 (b) 1095 (c) 1090 (d) 1097

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