Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 317 · 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 11? (a) 2890 (b) 2913 (c) 2896 (d) 2893
2. 2.What is the units digit of 8^20? (a) 6 (b) 4 (c) 8 (d) 2
3. 3.What is the remainder when 162 is divided by 11? (a) 9 (b) 8 (c) 3 (d) 7
4. 4.Which of these numbers is divisible by 3? (a) 2516 (b) 2511 (c) 2512 (d) 2510
5. 5.What is the remainder when 220 is divided by 8? (a) 6 (b) 3 (c) 4 (d) 5
6. 6.What is the units digit of 8^17? (a) 2 (b) 6 (c) 8 (d) 4
7. 7.What is the smallest number you must add to 57 to make it divisible by 5? (a) 2 (b) 3 (c) 1 (d) 4
8. 8.Which of these numbers is divisible by 11? (a) 5188 (b) 5204 (c) 5192 (d) 5200
9. 9.Which of these numbers is divisible by 11? (a) 9025 (b) 9038 (c) 9020 (d) 9016
10. 10.Which of these numbers is divisible by 9? (a) 1489 (b) 1510 (c) 1498 (d) 1494
11. 11.What is the smallest number you must add to 152 to make it divisible by 10? (a) 7 (b) 2 (c) 9 (d) 8
12. 12.Which of these numbers is divisible by 8? (a) 2049 (b) 2059 (c) 2047 (d) 2048
13. 13.Which of these numbers is divisible by 8? (a) 5692 (b) 5686 (c) 5677 (d) 5680
14. 14.What is the remainder when 367 is divided by 10? (a) 6 (b) 3 (c) 8 (d) 7
15. 15.What is the remainder when 698 is divided by 9? (a) 7 (b) 5 (c) 6 (d) 4
16. 16.What is the smallest number you must add to 325 to make it divisible by 6? (a) 5 (b) 3 (c) 1 (d) 4
17. 17.What is the remainder when 146 is divided by 3? (a) 0 (b) 2 (c) 1
18. 18.What is the smallest number you must add to 194 to make it divisible by 4? (a) 1 (b) 3 (c) 2
19. 19.What is the remainder when 132 is divided by 7? (a) 5 (b) 4 (c) 1 (d) 6
20. 20.What is the remainder when 242 is divided by 8? (a) 1 (b) 3 (c) 2 (d) 6
21. 21.What is the smallest number you must add to 305 to make it divisible by 9? (a) 2 (b) 1 (c) 8 (d) 3
22. 22.What is the remainder when 149 is divided by 5? (a) 1 (b) 3 (c) 2 (d) 4
23. 23.Which of these numbers is divisible by 4? (a) 3389 (b) 3395 (c) 3388 (d) 3387
24. 24.What is the remainder when 116 is divided by 9? (a) 8 (b) 6 (c) 7 (d) 1
25. 25.What is the units digit of 8^21? (a) 2 (b) 6 (c) 8 (d) 4
26. 26.What is the units digit of 9^15? (a) 1 (b) 8 (c) 9 (d) 7
27. 27.What is the units digit of 2^48? (a) 4 (b) 2 (c) 6 (d) 8
28. 28.Which of these numbers is divisible by 6? (a) 1603 (b) 1608 (c) 1616 (d) 1609
29. 29.What is the units digit of 3^29? (a) 7 (b) 3 (c) 9 (d) 1
30. 30.What is the units digit of 7^26? (a) 3 (b) 9 (c) 7 (d) 1
31. 31.What is the remainder when 250 is divided by 7? (a) 2 (b) 6 (c) 5 (d) 4
32. 32.What is the smallest number you must add to 209 to make it divisible by 4? (a) 1 (b) 2 (c) 3
33. 33.What is the remainder when 422 is divided by 9? (a) 6 (b) 1 (c) 8 (d) 7
34. 34.What is the smallest number you must add to 681 to make it divisible by 11? (a) 10 (b) 3 (c) 2 (d) 1
35. 35.Which of these numbers is divisible by 9? (a) 2996 (b) 2987 (c) 2971 (d) 2979
36. 36.What is the units digit of 4^27? (a) 3 (b) 6 (c) 4 (d) 5
37. 37.What is the remainder when 285 is divided by 7? (a) 6 (b) 2 (c) 4 (d) 5
38. 38.What is the units digit of 2^28? (a) 4 (b) 2 (c) 8 (d) 6
39. 39.What is the smallest number you must add to 323 to make it divisible by 7? (a) 6 (b) 4 (c) 5 (d) 1
40. 40.What is the remainder when 435 is divided by 10? (a) 4 (b) 6 (c) 5 (d) 7

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