Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 934 · 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 326 is divided by 12? (a) 1 (b) 3 (c) 10 (d) 2
2. 2.What is the units digit of 3^13? (a) 7 (b) 9 (c) 3 (d) 1
3. 3.Which of these numbers is divisible by 3? (a) 1235 (b) 1238 (c) 1241 (d) 1236
4. 4.What is the remainder when 689 is divided by 11? (a) 7 (b) 8 (c) 4 (d) 6
5. 5.What is the units digit of 3^20? (a) 9 (b) 3 (c) 1 (d) 7
6. 6.Which of these numbers is divisible by 11? (a) 1210 (b) 1216 (c) 1229 (d) 1201
7. 7.What is the remainder when 254 is divided by 4? (a) 3 (b) 2 (c) 1 (d) 0
8. 8.What is the smallest number you must add to 558 to make it divisible by 11? (a) 8 (b) 3 (c) 2 (d) 4
9. 9.What is the remainder when 141 is divided by 4? (a) 3 (b) 2 (c) 0 (d) 1
10. 10.What is the units digit of 3^10? (a) 7 (b) 1 (c) 3 (d) 9
11. 11.Which of these numbers is divisible by 11? (a) 750 (b) 766 (c) 747 (d) 748
12. 12.Which of these numbers is divisible by 4? (a) 1424 (b) 1422 (c) 1431 (d) 1425
13. 13.Which of these numbers is divisible by 11? (a) 3310 (b) 3296 (c) 3320 (d) 3300
14. 14.What is the units digit of 8^52? (a) 8 (b) 2 (c) 4 (d) 6
15. 15.What is the units digit of 9^16? (a) 9 (b) 1 (c) 0 (d) 2
16. 16.What is the units digit of 3^56? (a) 3 (b) 7 (c) 9 (d) 1
17. 17.Which of these numbers is divisible by 6? (a) 2264 (b) 2275 (c) 2272 (d) 2268
18. 18.What is the remainder when 399 is divided by 9? (a) 2 (b) 3 (c) 4 (d) 6
19. 19.What is the smallest number you must add to 516 to make it divisible by 10? (a) 5 (b) 6 (c) 4 (d) 3
20. 20.What is the remainder when 820 is divided by 11? (a) 7 (b) 8 (c) 5 (d) 6
21. 21.What is the smallest number you must add to 123 to make it divisible by 5? (a) 1 (b) 2 (c) 4 (d) 3
22. 22.What is the smallest number you must add to 188 to make it divisible by 7? (a) 6 (b) 3 (c) 2 (d) 1
23. 23.What is the remainder when 374 is divided by 5? (a) 2 (b) 4 (c) 3 (d) 1
24. 24.What is the smallest number you must add to 452 to make it divisible by 9? (a) 8 (b) 6 (c) 2 (d) 7
25. 25.What is the smallest number you must add to 241 to make it divisible by 4? (a) 3 (b) 2 (c) 1
26. 26.Which of these numbers is divisible by 3? (a) 861 (b) 860 (c) 866 (d) 862
27. 27.What is the units digit of 4^24? (a) 7 (b) 4 (c) 6 (d) 5
28. 28.What is the remainder when 230 is divided by 7? (a) 6 (b) 5 (c) 4 (d) 1
29. 29.Which of these numbers is divisible by 8? (a) 6503 (b) 6488 (c) 6481 (d) 6492
30. 30.What is the smallest number you must add to 623 to make it divisible by 9? (a) 2 (b) 8 (c) 7 (d) 6
31. 31.Which of these numbers is divisible by 3? (a) 241 (b) 245 (c) 247 (d) 243
32. 32.Which of these numbers is divisible by 8? (a) 367 (b) 352 (c) 355 (d) 348
33. 33.What is the remainder when 206 is divided by 11? (a) 7 (b) 9 (c) 3 (d) 8
34. 34.What is the smallest number you must add to 224 to make it divisible by 5? (a) 2 (b) 4 (c) 3 (d) 1
35. 35.What is the remainder when 79 is divided by 7? (a) 5 (b) 3 (c) 2 (d) 1
36. 36.What is the remainder when 291 is divided by 6? (a) 3 (b) 4 (c) 2 (d) 5
37. 37.Which of these numbers is divisible by 9? (a) 1028 (b) 1026 (c) 1023 (d) 1043
38. 38.What is the smallest number you must add to 239 to make it divisible by 3? (a) 1 (b) 2
39. 39.What is the smallest number you must add to 230 to make it divisible by 11? (a) 3 (b) 10 (c) 2 (d) 1
40. 40.What is the smallest number you must add to 367 to make it divisible by 11? (a) 4 (b) 6 (c) 8 (d) 7

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