Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 486 · 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 6? (a) 1024 (b) 1027 (c) 1018 (d) 1020
2. 2.Which of these numbers is divisible by 9? (a) 2628 (b) 2641 (c) 2629 (d) 2624
3. 3.What is the remainder when 492 is divided by 10? (a) 1 (b) 2 (c) 3 (d) 8
4. 4.What is the smallest number you must add to 232 to make it divisible by 3? (a) 1 (b) 2
5. 5.What is the remainder when 499 is divided by 11? (a) 4 (b) 7 (c) 5 (d) 3
6. 6.What is the smallest number you must add to 587 to make it divisible by 9? (a) 6 (b) 7 (c) 2 (d) 8
7. 7.What is the remainder when 397 is divided by 9? (a) 2 (b) 0 (c) 1 (d) 8
8. 8.What is the units digit of 8^10? (a) 2 (b) 6 (c) 4 (d) 8
9. 9.What is the smallest number you must add to 97 to make it divisible by 3? (a) 1 (b) 2
10. 10.What is the smallest number you must add to 117 to make it divisible by 8? (a) 4 (b) 2 (c) 5 (d) 3
11. 11.What is the smallest number you must add to 248 to make it divisible by 5? (a) 4 (b) 1 (c) 2 (d) 3
12. 12.Which of these numbers is divisible by 9? (a) 2946 (b) 2964 (c) 2952 (d) 2954
13. 13.What is the units digit of 3^55? (a) 7 (b) 3 (c) 1 (d) 9
14. 14.What is the units digit of 3^27? (a) 3 (b) 9 (c) 7 (d) 1
15. 15.Which of these numbers is divisible by 3? (a) 743 (b) 736 (c) 738 (d) 739
16. 16.What is the smallest number you must add to 554 to make it divisible by 8? (a) 2 (b) 5 (c) 6 (d) 7
17. 17.Which of these numbers is divisible by 6? (a) 2997 (b) 2993 (c) 2988 (d) 2986
18. 18.What is the remainder when 42 is divided by 4? (a) 0 (b) 2 (c) 1 (d) 3
19. 19.What is the units digit of 9^58? (a) 1 (b) 9 (c) 0 (d) 2
20. 20.What is the smallest number you must add to 179 to make it divisible by 4? (a) 1 (b) 2 (c) 3
21. 21.Which of these numbers is divisible by 8? (a) 6295 (b) 6298 (c) 6287 (d) 6288
22. 22.What is the smallest number you must add to 323 to make it divisible by 6? (a) 3 (b) 5 (c) 2 (d) 1
23. 23.Which of these numbers is divisible by 9? (a) 2271 (b) 2292 (c) 2277 (d) 2285
24. 24.What is the remainder when 427 is divided by 12? (a) 7 (b) 8 (c) 6 (d) 5
25. 25.Which of these numbers is divisible by 8? (a) 959 (b) 964 (c) 952 (d) 947
26. 26.What is the smallest number you must add to 138 to make it divisible by 7? (a) 3 (b) 5 (c) 2 (d) 1
27. 27.Which of these numbers is divisible by 8? (a) 6874 (b) 6881 (c) 6880 (d) 6891
28. 28.What is the smallest number you must add to 91 to make it divisible by 5? (a) 3 (b) 2 (c) 4 (d) 1
29. 29.What is the remainder when 461 is divided by 8? (a) 6 (b) 5 (c) 3 (d) 4
30. 30.What is the remainder when 671 is divided by 10? (a) 9 (b) 1 (c) 2 (d) 0
31. 31.What is the units digit of 9^40? (a) 0 (b) 2 (c) 9 (d) 1
32. 32.What is the units digit of 3^48? (a) 7 (b) 3 (c) 9 (d) 1
33. 33.Which of these numbers is divisible by 6? (a) 3589 (b) 3605 (c) 3597 (d) 3594
34. 34.What is the units digit of 2^10? (a) 8 (b) 2 (c) 6 (d) 4
35. 35.What is the smallest number you must add to 405 to make it divisible by 7? (a) 1 (b) 2 (c) 6 (d) 3
36. 36.What is the remainder when 200 is divided by 3? (a) 1 (b) 2 (c) 0
37. 37.What is the units digit of 3^24? (a) 3 (b) 1 (c) 7 (d) 9
38. 38.What is the smallest number you must add to 169 to make it divisible by 5? (a) 2 (b) 1 (c) 4 (d) 3
39. 39.What is the units digit of 4^15? (a) 6 (b) 4 (c) 3 (d) 5
40. 40.What is the smallest number you must add to 123 to make it divisible by 4? (a) 1 (b) 3 (c) 2

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