Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 132 · 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 smallest number you must add to 332 to make it divisible by 5? (a) 4 (b) 1 (c) 3 (d) 2
2. 2.What is the smallest number you must add to 302 to make it divisible by 5? (a) 2 (b) 1 (c) 3 (d) 4
3. 3.What is the smallest number you must add to 808 to make it divisible by 10? (a) 8 (b) 2 (c) 3 (d) 1
4. 4.Which of these numbers is divisible by 4? (a) 586 (b) 594 (c) 588 (d) 589
5. 5.Which of these numbers is divisible by 6? (a) 2986 (b) 2990 (c) 2982 (d) 2980
6. 6.Which of these numbers is divisible by 8? (a) 525 (b) 534 (c) 528 (d) 543
7. 7.Which of these numbers is divisible by 8? (a) 834 (b) 829 (c) 821 (d) 824
8. 8.What is the smallest number you must add to 483 to make it divisible by 10? (a) 6 (b) 3 (c) 7 (d) 8
9. 9.What is the smallest number you must add to 76 to make it divisible by 6? (a) 4 (b) 3 (c) 2 (d) 1
10. 10.What is the remainder when 205 is divided by 10? (a) 4 (b) 5 (c) 7 (d) 6
11. 11.What is the remainder when 655 is divided by 10? (a) 7 (b) 5 (c) 6 (d) 4
12. 12.Which of these numbers is divisible by 9? (a) 1514 (b) 1536 (c) 1528 (d) 1521
13. 13.What is the remainder when 162 is divided by 4? (a) 1 (b) 0 (c) 3 (d) 2
14. 14.What is the remainder when 346 is divided by 10? (a) 4 (b) 6 (c) 7 (d) 5
15. 15.What is the units digit of 8^18? (a) 8 (b) 6 (c) 4 (d) 2
16. 16.What is the remainder when 346 is divided by 8? (a) 1 (b) 3 (c) 2 (d) 6
17. 17.What is the remainder when 405 is divided by 6? (a) 2 (b) 4 (c) 5 (d) 3
18. 18.What is the smallest number you must add to 579 to make it divisible by 12? (a) 10 (b) 9 (c) 8 (d) 3
19. 19.Which of these numbers is divisible by 4? (a) 3413 (b) 3422 (c) 3416 (d) 3419
20. 20.Which of these numbers is divisible by 9? (a) 1027 (b) 1047 (c) 1039 (d) 1035
21. 21.What is the remainder when 363 is divided by 7? (a) 1 (b) 4 (c) 6 (d) 5
22. 22.What is the units digit of 9^20? (a) 9 (b) 2 (c) 1 (d) 0
23. 23.What is the units digit of 8^11? (a) 8 (b) 6 (c) 2 (d) 4
24. 24.What is the smallest number you must add to 795 to make it divisible by 12? (a) 8 (b) 3 (c) 9 (d) 10
25. 25.Which of these numbers is divisible by 9? (a) 4050 (b) 4051 (c) 4047 (d) 4063
26. 26.Which of these numbers is divisible by 6? (a) 623 (b) 615 (c) 626 (d) 618
27. 27.What is the remainder when 567 is divided by 8? (a) 6 (b) 5 (c) 7 (d) 1
28. 28.What is the units digit of 2^13? (a) 8 (b) 6 (c) 4 (d) 2
29. 29.What is the units digit of 4^33? (a) 3 (b) 6 (c) 5 (d) 4
30. 30.Which of these numbers is divisible by 11? (a) 2830 (b) 2844 (c) 2856 (d) 2838
31. 31.Which of these numbers is divisible by 9? (a) 5837 (b) 5820 (c) 5823 (d) 5831
32. 32.What is the smallest number you must add to 272 to make it divisible by 12? (a) 3 (b) 4 (c) 8 (d) 5
33. 33.Which of these numbers is divisible by 3? (a) 1140 (b) 1144 (c) 1139 (d) 1142
34. 34.What is the smallest number you must add to 452 to make it divisible by 12? (a) 4 (b) 3 (c) 5 (d) 8
35. 35.What is the units digit of 2^23? (a) 2 (b) 6 (c) 4 (d) 8
36. 36.What is the remainder when 155 is divided by 3? (a) 2 (b) 0 (c) 1
37. 37.What is the smallest number you must add to 309 to make it divisible by 4? (a) 1 (b) 2 (c) 3
38. 38.What is the units digit of 4^30? (a) 6 (b) 7 (c) 5 (d) 4
39. 39.What is the remainder when 83 is divided by 6? (a) 1 (b) 5 (c) 3 (d) 4
40. 40.Which of these numbers is divisible by 3? (a) 1395 (b) 1397 (c) 1393 (d) 1400

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