Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 242 · 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 units digit of 8^13? (a) 4 (b) 8 (c) 6 (d) 2
2. 2.What is the smallest number you must add to 484 to make it divisible by 7? (a) 6 (b) 4 (c) 1 (d) 5
3. 3.What is the smallest number you must add to 190 to make it divisible by 4? (a) 1 (b) 2 (c) 3
4. 4.What is the units digit of 8^27? (a) 6 (b) 4 (c) 2 (d) 8
5. 5.Which of these numbers is divisible by 11? (a) 7799 (b) 7818 (c) 7804 (d) 7793
6. 6.Which of these numbers is divisible by 8? (a) 5640 (b) 5649 (c) 5638 (d) 5644
7. 7.What is the smallest number you must add to 177 to make it divisible by 4? (a) 3 (b) 1 (c) 2
8. 8.Which of these numbers is divisible by 4? (a) 2155 (b) 2152 (c) 2150 (d) 2159
9. 9.What is the remainder when 227 is divided by 4? (a) 1 (b) 3 (c) 0 (d) 2
10. 10.Which of these numbers is divisible by 4? (a) 727 (b) 731 (c) 724 (d) 722
11. 11.What is the units digit of 7^56? (a) 3 (b) 1 (c) 7 (d) 9
12. 12.What is the remainder when 637 is divided by 8? (a) 3 (b) 5 (c) 4 (d) 6
13. 13.What is the smallest number you must add to 40 to make it divisible by 3? (a) 1 (b) 2
14. 14.Which of these numbers is divisible by 9? (a) 3612 (b) 3621 (c) 3605 (d) 3609
15. 15.Which of these numbers is divisible by 11? (a) 689 (b) 706 (c) 703 (d) 693
16. 16.What is the units digit of 3^37? (a) 1 (b) 3 (c) 7 (d) 9
17. 17.What is the units digit of 9^14? (a) 9 (b) 1 (c) 2 (d) 0
18. 18.What is the remainder when 438 is divided by 7? (a) 3 (b) 6 (c) 5 (d) 4
19. 19.What is the smallest number you must add to 489 to make it divisible by 8? (a) 5 (b) 1 (c) 6 (d) 7
20. 20.What is the smallest number you must add to 371 to make it divisible by 8? (a) 5 (b) 4 (c) 6 (d) 3
21. 21.Which of these numbers is divisible by 6? (a) 1771 (b) 1766 (c) 1763 (d) 1764
22. 22.What is the remainder when 122 is divided by 12? (a) 1 (b) 10 (c) 3 (d) 2
23. 23.What is the remainder when 263 is divided by 5? (a) 1 (b) 3 (c) 4 (d) 2
24. 24.What is the units digit of 9^25? (a) 8 (b) 1 (c) 7 (d) 9
25. 25.What is the remainder when 163 is divided by 7? (a) 1 (b) 3 (c) 2 (d) 5
26. 26.Which of these numbers is divisible by 3? (a) 1930 (b) 1924 (c) 1926 (d) 1927
27. 27.What is the remainder when 173 is divided by 9? (a) 3 (b) 7 (c) 1 (d) 2
28. 28.What is the units digit of 7^14? (a) 9 (b) 1 (c) 7 (d) 3
29. 29.Which of these numbers is divisible by 3? (a) 2630 (b) 2636 (c) 2633 (d) 2631
30. 30.What is the remainder when 704 is divided by 10? (a) 3 (b) 6 (c) 5 (d) 4
31. 31.What is the smallest number you must add to 446 to make it divisible by 6? (a) 3 (b) 4 (c) 2 (d) 5
32. 32.What is the units digit of 9^44? (a) 2 (b) 1 (c) 9 (d) 0
33. 33.What is the units digit of 3^48? (a) 3 (b) 7 (c) 1 (d) 9
34. 34.What is the remainder when 231 is divided by 12? (a) 9 (b) 4 (c) 3 (d) 2
35. 35.What is the remainder when 285 is divided by 10? (a) 7 (b) 4 (c) 6 (d) 5
36. 36.What is the remainder when 248 is divided by 6? (a) 3 (b) 1 (c) 2 (d) 4
37. 37.What is the units digit of 4^23? (a) 6 (b) 4 (c) 5 (d) 3
38. 38.What is the units digit of 7^13? (a) 3 (b) 9 (c) 1 (d) 7
39. 39.Which of these numbers is divisible by 3? (a) 259 (b) 257 (c) 262 (d) 258
40. 40.What is the units digit of 8^51? (a) 6 (b) 4 (c) 2 (d) 8

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