Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 945 · 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) 5155 (b) 5160 (c) 5172 (d) 5167
2. 2.Which of these numbers is divisible by 11? (a) 6020 (b) 6045 (c) 6034 (d) 6028
3. 3.Which of these numbers is divisible by 6? (a) 843 (b) 830 (c) 834 (d) 839
4. 4.What is the smallest number you must add to 674 to make it divisible by 10? (a) 4 (b) 6 (c) 5 (d) 7
5. 5.What is the remainder when 226 is divided by 6? (a) 2 (b) 3 (c) 4 (d) 5
6. 6.What is the remainder when 255 is divided by 6? (a) 4 (b) 3 (c) 2 (d) 5
7. 7.What is the units digit of 3^30? (a) 9 (b) 1 (c) 7 (d) 3
8. 8.Which of these numbers is divisible by 11? (a) 5900 (b) 5896 (c) 5916 (d) 5894
9. 9.What is the units digit of 3^47? (a) 3 (b) 7 (c) 9 (d) 1
10. 10.What is the remainder when 517 is divided by 9? (a) 6 (b) 3 (c) 4 (d) 5
11. 11.What is the remainder when 610 is divided by 8? (a) 6 (b) 2 (c) 1 (d) 3
12. 12.What is the units digit of 2^17? (a) 8 (b) 4 (c) 2 (d) 6
13. 13.Which of these numbers is divisible by 6? (a) 3340 (b) 3330 (c) 3334 (d) 3329
14. 14.What is the smallest number you must add to 487 to make it divisible by 7? (a) 5 (b) 4 (c) 3 (d) 2
15. 15.What is the units digit of 8^47? (a) 2 (b) 6 (c) 8 (d) 4
16. 16.What is the units digit of 4^15? (a) 4 (b) 5 (c) 6 (d) 3
17. 17.What is the remainder when 698 is divided by 10? (a) 8 (b) 7 (c) 2 (d) 9
18. 18.What is the remainder when 53 is divided by 4? (a) 2 (b) 1 (c) 0 (d) 3
19. 19.What is the smallest number you must add to 397 to make it divisible by 9? (a) 1 (b) 7 (c) 6 (d) 8
20. 20.What is the remainder when 512 is divided by 10? (a) 2 (b) 8 (c) 3 (d) 1
21. 21.What is the remainder when 141 is divided by 12? (a) 9 (b) 3 (c) 10 (d) 8
22. 22.What is the smallest number you must add to 458 to make it divisible by 8? (a) 6 (b) 2 (c) 5 (d) 7
23. 23.Which of these numbers is divisible by 9? (a) 4817 (b) 4831 (c) 4813 (d) 4815
24. 24.What is the remainder when 65 is divided by 4? (a) 3 (b) 0 (c) 2 (d) 1
25. 25.Which of these numbers is divisible by 3? (a) 2015 (b) 2018 (c) 2013 (d) 2011
26. 26.Which of these numbers is divisible by 6? (a) 2668 (b) 2659 (c) 2671 (d) 2664
27. 27.What is the smallest number you must add to 204 to make it divisible by 9? (a) 2 (b) 4 (c) 3 (d) 6
28. 28.What is the units digit of 3^49? (a) 3 (b) 7 (c) 9 (d) 1
29. 29.What is the remainder when 163 is divided by 3? (a) 2 (b) 0 (c) 1
30. 30.What is the remainder when 266 is divided by 4? (a) 1 (b) 0 (c) 2 (d) 3
31. 31.What is the units digit of 3^24? (a) 1 (b) 3 (c) 9 (d) 7
32. 32.What is the remainder when 294 is divided by 10? (a) 6 (b) 5 (c) 4 (d) 3
33. 33.What is the remainder when 167 is divided by 9? (a) 5 (b) 6 (c) 7 (d) 4
34. 34.What is the units digit of 8^13? (a) 6 (b) 2 (c) 8 (d) 4
35. 35.What is the remainder when 169 is divided by 3? (a) 2 (b) 1 (c) 0
36. 36.What is the units digit of 2^23? (a) 4 (b) 2 (c) 8 (d) 6
37. 37.What is the remainder when 722 is divided by 9? (a) 1 (b) 7 (c) 2 (d) 3
38. 38.Which of these numbers is divisible by 11? (a) 9173 (b) 9174 (c) 9195 (d) 9183
39. 39.What is the remainder when 340 is divided by 12? (a) 5 (b) 8 (c) 3 (d) 4
40. 40.What is the units digit of 8^31? (a) 2 (b) 8 (c) 6 (d) 4

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