Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 83 · 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 units digit of 3^32? (a) 3 (b) 7 (c) 1 (d) 9
2. 2.What is the smallest number you must add to 133 to make it divisible by 6? (a) 3 (b) 5 (c) 1 (d) 4
3. 3.What is the remainder when 179 is divided by 3? (a) 1 (b) 0 (c) 2
4. 4.What is the units digit of 3^36? (a) 9 (b) 1 (c) 7 (d) 3
5. 5.What is the remainder when 220 is divided by 3? (a) 0 (b) 1 (c) 2
6. 6.What is the smallest number you must add to 188 to make it divisible by 7? (a) 6 (b) 1 (c) 2 (d) 3
7. 7.Which of these numbers is divisible by 4? (a) 3042 (b) 3050 (c) 3047 (d) 3044
8. 8.What is the remainder when 488 is divided by 12? (a) 4 (b) 9 (c) 7 (d) 8
9. 9.Which of these numbers is divisible by 3? (a) 2451 (b) 2456 (c) 2449 (d) 2452
10. 10.What is the smallest number you must add to 61 to make it divisible by 4? (a) 3 (b) 2 (c) 1
11. 11.What is the remainder when 181 is divided by 7? (a) 1 (b) 4 (c) 5 (d) 6
12. 12.What is the smallest number you must add to 372 to make it divisible by 11? (a) 2 (b) 9 (c) 3 (d) 1
13. 13.What is the smallest number you must add to 683 to make it divisible by 10? (a) 6 (b) 8 (c) 3 (d) 7
14. 14.What is the units digit of 9^36? (a) 0 (b) 2 (c) 1 (d) 9
15. 15.What is the smallest number you must add to 399 to make it divisible by 5? (a) 4 (b) 1 (c) 2 (d) 3
16. 16.Which of these numbers is divisible by 6? (a) 3552 (b) 3548 (c) 3556 (d) 3563
17. 17.Which of these numbers is divisible by 9? (a) 5522 (b) 5517 (c) 5528 (d) 5515
18. 18.What is the smallest number you must add to 883 to make it divisible by 11? (a) 3 (b) 7 (c) 8 (d) 9
19. 19.What is the units digit of 7^48? (a) 7 (b) 9 (c) 3 (d) 1
20. 20.What is the units digit of 8^50? (a) 6 (b) 8 (c) 2 (d) 4
21. 21.Which of these numbers is divisible by 4? (a) 1289 (b) 1283 (c) 1284 (d) 1286
22. 22.Which of these numbers is divisible by 3? (a) 180 (b) 181 (c) 185 (d) 178
23. 23.Which of these numbers is divisible by 9? (a) 1359 (b) 1360 (c) 1373 (d) 1357
24. 24.What is the remainder when 291 is divided by 7? (a) 4 (b) 5 (c) 3 (d) 6
25. 25.Which of these numbers is divisible by 4? (a) 2073 (b) 2071 (c) 2068 (d) 2067
26. 26.What is the units digit of 4^13? (a) 6 (b) 4 (c) 5 (d) 3
27. 27.What is the units digit of 9^49? (a) 9 (b) 1 (c) 8 (d) 7
28. 28.Which of these numbers is divisible by 3? (a) 149 (b) 150 (c) 152 (d) 154
29. 29.Which of these numbers is divisible by 3? (a) 1190 (b) 1191 (c) 1196 (d) 1192
30. 30.Which of these numbers is divisible by 8? (a) 3719 (b) 3712 (c) 3725 (d) 3709
31. 31.What is the smallest number you must add to 814 to make it divisible by 12? (a) 3 (b) 1 (c) 2 (d) 10
32. 32.What is the smallest number you must add to 164 to make it divisible by 9? (a) 2 (b) 7 (c) 8 (d) 6
33. 33.Which of these numbers is divisible by 3? (a) 903 (b) 901 (c) 904 (d) 908
34. 34.What is the units digit of 3^24? (a) 7 (b) 3 (c) 9 (d) 1
35. 35.What is the units digit of 4^36? (a) 5 (b) 7 (c) 4 (d) 6
36. 36.What is the units digit of 3^50? (a) 3 (b) 7 (c) 1 (d) 9
37. 37.What is the units digit of 7^44? (a) 9 (b) 7 (c) 3 (d) 1
38. 38.What is the remainder when 234 is divided by 8? (a) 3 (b) 6 (c) 2 (d) 1
39. 39.Which of these numbers is divisible by 8? (a) 6956 (b) 6944 (c) 6942 (d) 6945
40. 40.Which of these numbers is divisible by 4? (a) 476 (b) 473 (c) 477 (d) 482

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