Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 440 · 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 167 to make it divisible by 6? (a) 5 (b) 1 (c) 2 (d) 3
2. 2.Which of these numbers is divisible by 11? (a) 2472 (b) 2460 (c) 2484 (d) 2464
3. 3.What is the remainder when 676 is divided by 10? (a) 6 (b) 5 (c) 4 (d) 7
4. 4.What is the units digit of 9^35? (a) 9 (b) 8 (c) 1 (d) 7
5. 5.What is the smallest number you must add to 240 to make it divisible by 11? (a) 9 (b) 3 (c) 1 (d) 2
6. 6.What is the remainder when 80 is divided by 3? (a) 1 (b) 0 (c) 2
7. 7.Which of these numbers is divisible by 4? (a) 2215 (b) 2210 (c) 2219 (d) 2212
8. 8.What is the units digit of 2^20? (a) 8 (b) 2 (c) 6 (d) 4
9. 9.What is the smallest number you must add to 210 to make it divisible by 4? (a) 2 (b) 3 (c) 1
10. 10.Which of these numbers is divisible by 3? (a) 845 (b) 843 (c) 842 (d) 848
11. 11.Which of these numbers is divisible by 8? (a) 1528 (b) 1537 (c) 1530 (d) 1526
12. 12.What is the smallest number you must add to 727 to make it divisible by 12? (a) 4 (b) 7 (c) 6 (d) 5
13. 13.Which of these numbers is divisible by 3? (a) 1710 (b) 1711 (c) 1714 (d) 1708
14. 14.What is the smallest number you must add to 161 to make it divisible by 11? (a) 7 (b) 4 (c) 5 (d) 3
15. 15.What is the remainder when 173 is divided by 10? (a) 7 (b) 2 (c) 3 (d) 4
16. 16.Which of these numbers is divisible by 6? (a) 1123 (b) 1136 (c) 1133 (d) 1128
17. 17.What is the remainder when 333 is divided by 6? (a) 5 (b) 2 (c) 4 (d) 3
18. 18.What is the smallest number you must add to 120 to make it divisible by 9? (a) 7 (b) 5 (c) 6 (d) 3
19. 19.What is the smallest number you must add to 172 to make it divisible by 10? (a) 8 (b) 2 (c) 7 (d) 9
20. 20.What is the smallest number you must add to 580 to make it divisible by 11? (a) 4 (b) 3 (c) 2 (d) 8
21. 21.What is the remainder when 374 is divided by 6? (a) 3 (b) 2 (c) 4 (d) 1
22. 22.Which of these numbers is divisible by 9? (a) 842 (b) 828 (c) 826 (d) 830
23. 23.What is the remainder when 74 is divided by 5? (a) 1 (b) 2 (c) 3 (d) 4
24. 24.What is the units digit of 9^43? (a) 8 (b) 9 (c) 7 (d) 1
25. 25.What is the smallest number you must add to 156 to make it divisible by 7? (a) 2 (b) 4 (c) 5 (d) 6
26. 26.What is the units digit of 3^58? (a) 3 (b) 9 (c) 7 (d) 1
27. 27.What is the remainder when 93 is divided by 8? (a) 3 (b) 5 (c) 6 (d) 4
28. 28.Which of these numbers is divisible by 9? (a) 6067 (b) 6078 (c) 6085 (d) 6075
29. 29.What is the remainder when 148 is divided by 5? (a) 4 (b) 1 (c) 3 (d) 2
30. 30.Which of these numbers is divisible by 4? (a) 1094 (b) 1099 (c) 1096 (d) 1103
31. 31.Which of these numbers is divisible by 3? (a) 616 (b) 619 (c) 613 (d) 615
32. 32.Which of these numbers is divisible by 4? (a) 1656 (b) 1657 (c) 1661 (d) 1654
33. 33.What is the remainder when 538 is divided by 9? (a) 6 (b) 7 (c) 2 (d) 8
34. 34.Which of these numbers is divisible by 3? (a) 1073 (b) 1078 (c) 1074 (d) 1075
35. 35.What is the remainder when 748 is divided by 12? (a) 5 (b) 4 (c) 8 (d) 3
36. 36.Which of these numbers is divisible by 3? (a) 495 (b) 494 (c) 500 (d) 496
37. 37.What is the units digit of 7^16? (a) 9 (b) 1 (c) 3 (d) 7
38. 38.What is the units digit of 4^38? (a) 7 (b) 5 (c) 6 (d) 4
39. 39.Which of these numbers is divisible by 8? (a) 6055 (b) 6062 (c) 6056 (d) 6070
40. 40.What is the remainder when 181 is divided by 3? (a) 2 (b) 0 (c) 1

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