Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 232 · 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 7^18? (a) 3 (b) 9 (c) 7 (d) 1
2. 2.Which of these numbers is divisible by 3? (a) 906 (b) 911 (c) 908 (d) 905
3. 3.What is the remainder when 155 is divided by 10? (a) 6 (b) 7 (c) 4 (d) 5
4. 4.What is the smallest number you must add to 388 to make it divisible by 6? (a) 1 (b) 2 (c) 4 (d) 3
5. 5.Which of these numbers is divisible by 3? (a) 2071 (b) 2073 (c) 2074 (d) 2077
6. 6.What is the smallest number you must add to 171 to make it divisible by 11? (a) 4 (b) 6 (c) 5 (d) 7
7. 7.What is the smallest number you must add to 65 to make it divisible by 3? (a) 2 (b) 1
8. 8.Which of these numbers is divisible by 11? (a) 3868 (b) 3890 (c) 3872 (d) 3876
9. 9.What is the remainder when 97 is divided by 9? (a) 6 (b) 2 (c) 7 (d) 8
10. 10.What is the smallest number you must add to 283 to make it divisible by 12? (a) 5 (b) 7 (c) 4 (d) 6
11. 11.What is the remainder when 314 is divided by 11? (a) 6 (b) 5 (c) 8 (d) 7
12. 12.What is the smallest number you must add to 323 to make it divisible by 4? (a) 3 (b) 2 (c) 1
13. 13.Which of these numbers is divisible by 8? (a) 2020 (b) 2036 (c) 2026 (d) 2024
14. 14.What is the units digit of 8^46? (a) 6 (b) 4 (c) 8 (d) 2
15. 15.What is the remainder when 73 is divided by 7? (a) 5 (b) 3 (c) 4 (d) 2
16. 16.What is the units digit of 9^25? (a) 1 (b) 8 (c) 9 (d) 7
17. 17.What is the smallest number you must add to 236 to make it divisible by 12? (a) 5 (b) 3 (c) 4 (d) 8
18. 18.Which of these numbers is divisible by 8? (a) 6828 (b) 6837 (c) 6823 (d) 6824
19. 19.What is the units digit of 4^13? (a) 6 (b) 3 (c) 4 (d) 5
20. 20.What is the smallest number you must add to 289 to make it divisible by 4? (a) 1 (b) 3 (c) 2
21. 21.What is the units digit of 8^51? (a) 2 (b) 6 (c) 4 (d) 8
22. 22.What is the remainder when 158 is divided by 3? (a) 0 (b) 1 (c) 2
23. 23.Which of these numbers is divisible by 9? (a) 7958 (b) 7941 (c) 7951 (d) 7947
24. 24.What is the smallest number you must add to 344 to make it divisible by 5? (a) 1 (b) 3 (c) 4 (d) 2
25. 25.Which of these numbers is divisible by 9? (a) 702 (b) 719 (c) 710 (d) 698
26. 26.What is the remainder when 429 is divided by 7? (a) 1 (b) 2 (c) 5 (d) 3
27. 27.Which of these numbers is divisible by 6? (a) 1007 (b) 1009 (c) 1008 (d) 1015
28. 28.What is the remainder when 430 is divided by 9? (a) 2 (b) 8 (c) 6 (d) 7
29. 29.What is the units digit of 4^44? (a) 7 (b) 6 (c) 5 (d) 4
30. 30.Which of these numbers is divisible by 11? (a) 613 (b) 624 (c) 616 (d) 630
31. 31.What is the units digit of 7^30? (a) 7 (b) 9 (c) 1 (d) 3
32. 32.What is the smallest number you must add to 91 to make it divisible by 3? (a) 1 (b) 2
33. 33.What is the smallest number you must add to 633 to make it divisible by 8? (a) 5 (b) 7 (c) 1 (d) 6
34. 34.What is the units digit of 3^51? (a) 9 (b) 3 (c) 1 (d) 7
35. 35.What is the smallest number you must add to 149 to make it divisible by 9? (a) 3 (b) 6 (c) 4 (d) 5
36. 36.What is the units digit of 8^24? (a) 2 (b) 4 (c) 6 (d) 8
37. 37.What is the smallest number you must add to 828 to make it divisible by 11? (a) 9 (b) 8 (c) 7 (d) 3
38. 38.Which of these numbers is divisible by 4? (a) 445 (b) 454 (c) 451 (d) 448
39. 39.What is the units digit of 4^45? (a) 4 (b) 5 (c) 3 (d) 6
40. 40.What is the units digit of 8^43? (a) 6 (b) 8 (c) 2 (d) 4

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