Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 197 · 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 11? (a) 3423 (b) 3411 (c) 3410 (d) 3407
2. 2.What is the smallest number you must add to 518 to make it divisible by 9? (a) 3 (b) 4 (c) 6 (d) 5
3. 3.Which of these numbers is divisible by 9? (a) 6165 (b) 6157 (c) 6166 (d) 6179
4. 4.Which of these numbers is divisible by 6? (a) 4185 (b) 4195 (c) 4189 (d) 4188
5. 5.What is the units digit of 2^57? (a) 4 (b) 8 (c) 2 (d) 6
6. 6.Which of these numbers is divisible by 3? (a) 2651 (b) 2644 (c) 2648 (d) 2646
7. 7.What is the remainder when 853 is divided by 11? (a) 6 (b) 7 (c) 8 (d) 5
8. 8.What is the remainder when 128 is divided by 5? (a) 3 (b) 1 (c) 4 (d) 2
9. 9.What is the smallest number you must add to 295 to make it divisible by 12? (a) 6 (b) 4 (c) 7 (d) 5
10. 10.What is the smallest number you must add to 496 to make it divisible by 9? (a) 7 (b) 6 (c) 1 (d) 8
11. 11.What is the smallest number you must add to 547 to make it divisible by 9? (a) 3 (b) 2 (c) 7 (d) 1
12. 12.What is the remainder when 329 is divided by 6? (a) 4 (b) 1 (c) 3 (d) 5
13. 13.What is the units digit of 4^18? (a) 4 (b) 6 (c) 7 (d) 5
14. 14.What is the units digit of 3^29? (a) 3 (b) 1 (c) 7 (d) 9
15. 15.What is the smallest number you must add to 361 to make it divisible by 5? (a) 2 (b) 4 (c) 3 (d) 1
16. 16.What is the remainder when 157 is divided by 3? (a) 2 (b) 1 (c) 0
17. 17.What is the units digit of 8^28? (a) 6 (b) 4 (c) 2 (d) 8
18. 18.What is the remainder when 305 is divided by 10? (a) 4 (b) 6 (c) 7 (d) 5
19. 19.What is the remainder when 678 is divided by 12? (a) 8 (b) 7 (c) 5 (d) 6
20. 20.What is the remainder when 204 is divided by 10? (a) 4 (b) 6 (c) 3 (d) 5
21. 21.What is the remainder when 62 is divided by 6? (a) 3 (b) 2 (c) 4 (d) 1
22. 22.What is the remainder when 232 is divided by 9? (a) 7 (b) 6 (c) 8 (d) 2
23. 23.What is the units digit of 4^60? (a) 6 (b) 5 (c) 4 (d) 7
24. 24.What is the remainder when 82 is divided by 8? (a) 6 (b) 3 (c) 2 (d) 1
25. 25.What is the smallest number you must add to 321 to make it divisible by 10? (a) 7 (b) 9 (c) 8 (d) 1
26. 26.What is the units digit of 8^50? (a) 4 (b) 2 (c) 8 (d) 6
27. 27.What is the smallest number you must add to 85 to make it divisible by 4? (a) 2 (b) 1 (c) 3
28. 28.Which of these numbers is divisible by 6? (a) 4163 (b) 4158 (c) 4168 (d) 4155
29. 29.What is the units digit of 7^60? (a) 3 (b) 9 (c) 1 (d) 7
30. 30.Which of these numbers is divisible by 9? (a) 6046 (b) 6032 (c) 6026 (d) 6030
31. 31.Which of these numbers is divisible by 3? (a) 2590 (b) 2585 (c) 2588 (d) 2586
32. 32.What is the units digit of 2^52? (a) 2 (b) 4 (c) 6 (d) 8
33. 33.What is the smallest number you must add to 190 to make it divisible by 11? (a) 9 (b) 3 (c) 8 (d) 7
34. 34.What is the smallest number you must add to 95 to make it divisible by 4? (a) 2 (b) 3 (c) 1
35. 35.What is the units digit of 9^48? (a) 2 (b) 0 (c) 1 (d) 9
36. 36.Which of these numbers is divisible by 8? (a) 1020 (b) 1025 (c) 1024 (d) 1038
37. 37.What is the remainder when 77 is divided by 3? (a) 2 (b) 0 (c) 1
38. 38.What is the smallest number you must add to 427 to make it divisible by 9? (a) 4 (b) 6 (c) 5 (d) 7
39. 39.What is the smallest number you must add to 279 to make it divisible by 6? (a) 4 (b) 2 (c) 5 (d) 3
40. 40.Which of these numbers is divisible by 3? (a) 859 (b) 862 (c) 857 (d) 858

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