Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 43 · 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) 1705 (b) 1695 (c) 1699 (d) 1698
2. 2.Which of these numbers is divisible by 11? (a) 8421 (b) 8405 (c) 8434 (d) 8415
3. 3.What is the remainder when 695 is divided by 10? (a) 4 (b) 7 (c) 5 (d) 6
4. 4.What is the remainder when 164 is divided by 3? (a) 0 (b) 2 (c) 1
5. 5.Which of these numbers is divisible by 6? (a) 2599 (b) 2592 (c) 2591 (d) 2594
6. 6.What is the remainder when 225 is divided by 8? (a) 7 (b) 2 (c) 1 (d) 0
7. 7.What is the remainder when 470 is divided by 9? (a) 1 (b) 3 (c) 2 (d) 7
8. 8.What is the smallest number you must add to 53 to make it divisible by 3? (a) 1 (b) 2
9. 9.Which of these numbers is divisible by 4? (a) 2174 (b) 2178 (c) 2171 (d) 2172
10. 10.What is the units digit of 2^48? (a) 8 (b) 2 (c) 4 (d) 6
11. 11.Which of these numbers is divisible by 9? (a) 683 (b) 673 (c) 660 (d) 666
12. 12.What is the smallest number you must add to 178 to make it divisible by 10? (a) 3 (b) 8 (c) 1 (d) 2
13. 13.Which of these numbers is divisible by 6? (a) 4908 (b) 4916 (c) 4912 (d) 4903
14. 14.What is the smallest number you must add to 239 to make it divisible by 3? (a) 2 (b) 1
15. 15.Which of these numbers is divisible by 4? (a) 777 (b) 781 (c) 773 (d) 776
16. 16.What is the smallest number you must add to 552 to make it divisible by 7? (a) 3 (b) 2 (c) 6 (d) 1
17. 17.What is the smallest number you must add to 523 to make it divisible by 10? (a) 8 (b) 6 (c) 3 (d) 7
18. 18.What is the remainder when 241 is divided by 3? (a) 2 (b) 0 (c) 1
19. 19.What is the smallest number you must add to 595 to make it divisible by 10? (a) 7 (b) 6 (c) 5 (d) 4
20. 20.What is the units digit of 2^30? (a) 4 (b) 6 (c) 8 (d) 2
21. 21.What is the units digit of 9^51? (a) 7 (b) 8 (c) 9 (d) 1
22. 22.What is the remainder when 190 is divided by 4? (a) 3 (b) 0 (c) 2 (d) 1
23. 23.Which of these numbers is divisible by 6? (a) 4949 (b) 4943 (c) 4938 (d) 4933
24. 24.What is the remainder when 665 is divided by 12? (a) 4 (b) 6 (c) 5 (d) 7
25. 25.Which of these numbers is divisible by 8? (a) 6836 (b) 6817 (c) 6824 (d) 6825
26. 26.What is the smallest number you must add to 442 to make it divisible by 10? (a) 2 (b) 8 (c) 7 (d) 9
27. 27.What is the units digit of 7^26? (a) 7 (b) 1 (c) 9 (d) 3
28. 28.Which of these numbers is divisible by 6? (a) 4635 (b) 4628 (c) 4624 (d) 4626
29. 29.What is the units digit of 9^15? (a) 8 (b) 9 (c) 7 (d) 1
30. 30.What is the remainder when 138 is divided by 5? (a) 1 (b) 4 (c) 3 (d) 2
31. 31.What is the smallest number you must add to 302 to make it divisible by 4? (a) 2 (b) 3 (c) 1
32. 32.What is the remainder when 221 is divided by 8? (a) 3 (b) 6 (c) 5 (d) 4
33. 33.What is the remainder when 291 is divided by 6? (a) 2 (b) 4 (c) 5 (d) 3
34. 34.What is the smallest number you must add to 102 to make it divisible by 5? (a) 1 (b) 3 (c) 4 (d) 2
35. 35.What is the remainder when 277 is divided by 4? (a) 0 (b) 3 (c) 1 (d) 2
36. 36.Which of these numbers is divisible by 6? (a) 1202 (b) 1209 (c) 1196 (d) 1200
37. 37.What is the units digit of 9^13? (a) 9 (b) 7 (c) 8 (d) 1
38. 38.What is the smallest number you must add to 516 to make it divisible by 10? (a) 3 (b) 6 (c) 5 (d) 4
39. 39.What is the smallest number you must add to 117 to make it divisible by 5? (a) 4 (b) 3 (c) 2 (d) 1
40. 40.What is the units digit of 4^35? (a) 6 (b) 3 (c) 5 (d) 4

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