Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 614 · 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 164 is divided by 9? (a) 7 (b) 1 (c) 3 (d) 2
2. 2.What is the units digit of 8^41? (a) 4 (b) 6 (c) 2 (d) 8
3. 3.What is the remainder when 149 is divided by 6? (a) 4 (b) 5 (c) 3 (d) 1
4. 4.Which of these numbers is divisible by 6? (a) 5370 (b) 5377 (c) 5368 (d) 5372
5. 5.What is the units digit of 9^50? (a) 0 (b) 2 (c) 9 (d) 1
6. 6.What is the smallest number you must add to 91 to make it divisible by 3? (a) 2 (b) 1
7. 7.Which of these numbers is divisible by 9? (a) 4006 (b) 4025 (c) 4019 (d) 4014
8. 8.What is the remainder when 138 is divided by 9? (a) 4 (b) 6 (c) 2 (d) 3
9. 9.Which of these numbers is divisible by 4? (a) 2104 (b) 2101 (c) 2105 (d) 2110
10. 10.Which of these numbers is divisible by 9? (a) 2012 (b) 2031 (c) 2018 (d) 2016
11. 11.What is the units digit of 9^13? (a) 8 (b) 1 (c) 7 (d) 9
12. 12.What is the remainder when 45 is divided by 4? (a) 0 (b) 2 (c) 3 (d) 1
13. 13.What is the smallest number you must add to 197 to make it divisible by 3? (a) 2 (b) 1
14. 14.What is the remainder when 94 is divided by 3? (a) 1 (b) 0 (c) 2
15. 15.What is the remainder when 688 is divided by 11? (a) 5 (b) 7 (c) 8 (d) 6
16. 16.Which of these numbers is divisible by 3? (a) 648 (b) 646 (c) 652 (d) 650
17. 17.Which of these numbers is divisible by 9? (a) 3639 (b) 3651 (c) 3660 (d) 3645
18. 18.What is the units digit of 8^22? (a) 8 (b) 6 (c) 4 (d) 2
19. 19.Which of these numbers is divisible by 8? (a) 5179 (b) 5174 (c) 5166 (d) 5168
20. 20.What is the units digit of 3^23? (a) 3 (b) 7 (c) 9 (d) 1
21. 21.Which of these numbers is divisible by 8? (a) 6823 (b) 6834 (c) 6828 (d) 6824
22. 22.What is the remainder when 446 is divided by 7? (a) 6 (b) 4 (c) 2 (d) 5
23. 23.What is the units digit of 2^42? (a) 6 (b) 4 (c) 8 (d) 2
24. 24.What is the remainder when 489 is divided by 9? (a) 6 (b) 3 (c) 4 (d) 2
25. 25.Which of these numbers is divisible by 9? (a) 3628 (b) 3650 (c) 3636 (d) 3642
26. 26.What is the units digit of 8^28? (a) 4 (b) 6 (c) 8 (d) 2
27. 27.What is the remainder when 155 is divided by 7? (a) 1 (b) 6 (c) 0 (d) 2
28. 28.Which of these numbers is divisible by 11? (a) 893 (b) 891 (c) 905 (d) 888
29. 29.What is the smallest number you must add to 283 to make it divisible by 6? (a) 5 (b) 3 (c) 1 (d) 4
30. 30.What is the units digit of 2^22? (a) 2 (b) 6 (c) 4 (d) 8
31. 31.What is the remainder when 292 is divided by 5? (a) 3 (b) 4 (c) 1 (d) 2
32. 32.What is the remainder when 286 is divided by 7? (a) 4 (b) 1 (c) 6 (d) 5
33. 33.Which of these numbers is divisible by 4? (a) 1885 (b) 1888 (c) 1889 (d) 1893
34. 34.What is the smallest number you must add to 280 to make it divisible by 12? (a) 7 (b) 8 (c) 9 (d) 4
35. 35.What is the smallest number you must add to 494 to make it divisible by 11? (a) 10 (b) 2 (c) 3 (d) 1
36. 36.Which of these numbers is divisible by 6? (a) 3089 (b) 3081 (c) 3076 (d) 3078
37. 37.What is the smallest number you must add to 221 to make it divisible by 4? (a) 1 (b) 3 (c) 2
38. 38.Which of these numbers is divisible by 9? (a) 7587 (b) 7589 (c) 7599 (d) 7583
39. 39.What is the units digit of 7^16? (a) 1 (b) 3 (c) 9 (d) 7
40. 40.What is the units digit of 9^45? (a) 9 (b) 1 (c) 7 (d) 8

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