Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 386 · 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 remainder when 211 is divided by 6? (a) 1 (b) 0 (c) 2 (d) 5
2. 2.What is the units digit of 7^30? (a) 3 (b) 1 (c) 9 (d) 7
3. 3.Which of these numbers is divisible by 11? (a) 1164 (b) 1166 (c) 1187 (d) 1171
4. 4.Which of these numbers is divisible by 9? (a) 2089 (b) 2101 (c) 2088 (d) 2087
5. 5.What is the remainder when 547 is divided by 11? (a) 8 (b) 9 (c) 3 (d) 7
6. 6.What is the units digit of 9^54? (a) 2 (b) 0 (c) 9 (d) 1
7. 7.Which of these numbers is divisible by 9? (a) 2331 (b) 2333 (c) 2347 (d) 2326
8. 8.Which of these numbers is divisible by 11? (a) 4933 (b) 4952 (c) 4939 (d) 4940
9. 9.What is the smallest number you must add to 469 to make it divisible by 8? (a) 5 (b) 4 (c) 2 (d) 3
10. 10.What is the units digit of 9^17? (a) 8 (b) 9 (c) 1 (d) 7
11. 11.What is the smallest number you must add to 540 to make it divisible by 8? (a) 6 (b) 3 (c) 5 (d) 4
12. 12.What is the units digit of 9^58? (a) 9 (b) 1 (c) 2 (d) 0
13. 13.What is the units digit of 3^12? (a) 7 (b) 1 (c) 9 (d) 3
14. 14.Which of these numbers is divisible by 9? (a) 7913 (b) 7897 (c) 7908 (d) 7902
15. 15.What is the smallest number you must add to 406 to make it divisible by 9? (a) 8 (b) 7 (c) 1 (d) 6
16. 16.What is the units digit of 2^39? (a) 8 (b) 4 (c) 2 (d) 6
17. 17.Which of these numbers is divisible by 4? (a) 3049 (b) 3044 (c) 3047 (d) 3043
18. 18.What is the units digit of 4^43? (a) 3 (b) 6 (c) 4 (d) 5
19. 19.What is the units digit of 2^11? (a) 4 (b) 8 (c) 2 (d) 6
20. 20.What is the units digit of 7^36? (a) 7 (b) 1 (c) 3 (d) 9
21. 21.What is the remainder when 163 is divided by 4? (a) 2 (b) 0 (c) 1 (d) 3
22. 22.What is the remainder when 141 is divided by 12? (a) 8 (b) 10 (c) 3 (d) 9
23. 23.What is the units digit of 4^49? (a) 4 (b) 5 (c) 3 (d) 6
24. 24.What is the units digit of 8^29? (a) 4 (b) 6 (c) 8 (d) 2
25. 25.Which of these numbers is divisible by 9? (a) 4233 (b) 4239 (c) 4253 (d) 4246
26. 26.What is the smallest number you must add to 310 to make it divisible by 11? (a) 10 (b) 2 (c) 9 (d) 8
27. 27.What is the remainder when 549 is divided by 7? (a) 4 (b) 2 (c) 5 (d) 3
28. 28.What is the units digit of 4^22? (a) 6 (b) 7 (c) 5 (d) 4
29. 29.Which of these numbers is divisible by 11? (a) 4002 (b) 4008 (c) 4024 (d) 4004
30. 30.Which of these numbers is divisible by 4? (a) 887 (b) 884 (c) 882 (d) 889
31. 31.Which of these numbers is divisible by 6? (a) 616 (b) 612 (c) 607 (d) 620
32. 32.What is the smallest number you must add to 386 to make it divisible by 5? (a) 4 (b) 3 (c) 2 (d) 1
33. 33.What is the smallest number you must add to 87 to make it divisible by 5? (a) 1 (b) 2 (c) 3 (d) 4
34. 34.Which of these numbers is divisible by 8? (a) 446 (b) 438 (c) 453 (d) 440
35. 35.What is the units digit of 2^15? (a) 6 (b) 8 (c) 2 (d) 4
36. 36.What is the smallest number you must add to 459 to make it divisible by 6? (a) 3 (b) 2 (c) 4 (d) 5
37. 37.What is the units digit of 9^12? (a) 1 (b) 9 (c) 2 (d) 0
38. 38.What is the remainder when 100 is divided by 3? (a) 1 (b) 0 (c) 2
39. 39.Which of these numbers is divisible by 11? (a) 9372 (b) 9364 (c) 9380 (d) 9385
40. 40.What is the smallest number you must add to 353 to make it divisible by 9? (a) 8 (b) 2 (c) 7 (d) 6

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