Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 612 · 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 177 is divided by 4? (a) 3 (b) 0 (c) 2 (d) 1
2. 2.What is the remainder when 97 is divided by 3? (a) 0 (b) 2 (c) 1
3. 3.What is the remainder when 121 is divided by 3? (a) 0 (b) 1 (c) 2
4. 4.What is the remainder when 559 is divided by 9? (a) 2 (b) 1 (c) 8 (d) 0
5. 5.Which of these numbers is divisible by 6? (a) 2784 (b) 2793 (c) 2788 (d) 2782
6. 6.What is the smallest number you must add to 349 to make it divisible by 6? (a) 3 (b) 5 (c) 4 (d) 1
7. 7.What is the smallest number you must add to 597 to make it divisible by 9? (a) 3 (b) 7 (c) 5 (d) 6
8. 8.What is the remainder when 152 is divided by 6? (a) 2 (b) 4 (c) 3 (d) 1
9. 9.Which of these numbers is divisible by 8? (a) 6373 (b) 6362 (c) 6360 (d) 6354
10. 10.What is the smallest number you must add to 164 to make it divisible by 3? (a) 2 (b) 1
11. 11.What is the units digit of 2^15? (a) 6 (b) 8 (c) 2 (d) 4
12. 12.What is the units digit of 3^37? (a) 3 (b) 1 (c) 7 (d) 9
13. 13.Which of these numbers is divisible by 6? (a) 4488 (b) 4493 (c) 4496 (d) 4486
14. 14.What is the remainder when 441 is divided by 11? (a) 1 (b) 2 (c) 10 (d) 0
15. 15.What is the smallest number you must add to 331 to make it divisible by 5? (a) 3 (b) 4 (c) 2 (d) 1
16. 16.What is the smallest number you must add to 40 to make it divisible by 3? (a) 2 (b) 1
17. 17.What is the units digit of 2^33? (a) 8 (b) 4 (c) 6 (d) 2
18. 18.What is the smallest number you must add to 86 to make it divisible by 5? (a) 1 (b) 4 (c) 3 (d) 2
19. 19.What is the smallest number you must add to 104 to make it divisible by 5? (a) 1 (b) 4 (c) 2 (d) 3
20. 20.Which of these numbers is divisible by 3? (a) 362 (b) 357 (c) 359 (d) 355
21. 21.Which of these numbers is divisible by 9? (a) 5949 (b) 5947 (c) 5950 (d) 5965
22. 22.What is the units digit of 9^48? (a) 1 (b) 2 (c) 0 (d) 9
23. 23.Which of these numbers is divisible by 6? (a) 4973 (b) 4976 (c) 4968 (d) 4967
24. 24.Which of these numbers is divisible by 4? (a) 2059 (b) 2061 (c) 2060 (d) 2067
25. 25.Which of these numbers is divisible by 9? (a) 2042 (b) 2025 (c) 2027 (d) 2023
26. 26.What is the units digit of 9^38? (a) 2 (b) 9 (c) 0 (d) 1
27. 27.What is the remainder when 505 is divided by 8? (a) 0 (b) 7 (c) 1 (d) 2
28. 28.What is the units digit of 7^14? (a) 1 (b) 7 (c) 3 (d) 9
29. 29.What is the remainder when 191 is divided by 7? (a) 2 (b) 3 (c) 1 (d) 5
30. 30.Which of these numbers is divisible by 9? (a) 3030 (b) 3036 (c) 3024 (d) 3017
31. 31.Which of these numbers is divisible by 6? (a) 2811 (b) 2817 (c) 2804 (d) 2808
32. 32.Which of these numbers is divisible by 6? (a) 3993 (b) 4003 (c) 4001 (d) 3996
33. 33.What is the units digit of 7^33? (a) 7 (b) 9 (c) 3 (d) 1
34. 34.What is the units digit of 8^34? (a) 2 (b) 4 (c) 8 (d) 6
35. 35.What is the smallest number you must add to 316 to make it divisible by 6? (a) 4 (b) 2 (c) 3 (d) 1
36. 36.What is the smallest number you must add to 226 to make it divisible by 6? (a) 4 (b) 3 (c) 2 (d) 1
37. 37.Which of these numbers is divisible by 11? (a) 8569 (b) 8579 (c) 8587 (d) 8566
38. 38.What is the remainder when 226 is divided by 6? (a) 5 (b) 4 (c) 2 (d) 3
39. 39.What is the remainder when 462 is divided by 8? (a) 7 (b) 2 (c) 5 (d) 6
40. 40.What is the remainder when 94 is divided by 6? (a) 2 (b) 5 (c) 4 (d) 3

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