Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 503 · 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 7^28? (a) 9 (b) 1 (c) 3 (d) 7
2. 2.What is the remainder when 229 is divided by 11? (a) 9 (b) 2 (c) 10 (d) 8
3. 3.What is the smallest number you must add to 230 to make it divisible by 4? (a) 1 (b) 3 (c) 2
4. 4.Which of these numbers is divisible by 11? (a) 7943 (b) 7939 (c) 7963 (d) 7942
5. 5.What is the units digit of 8^45? (a) 6 (b) 2 (c) 8 (d) 4
6. 6.What is the remainder when 81 is divided by 4? (a) 3 (b) 2 (c) 1 (d) 0
7. 7.What is the units digit of 4^38? (a) 4 (b) 7 (c) 6 (d) 5
8. 8.Which of these numbers is divisible by 9? (a) 918 (b) 928 (c) 913 (d) 923
9. 9.What is the smallest number you must add to 614 to make it divisible by 12? (a) 2 (b) 10 (c) 11 (d) 9
10. 10.Which of these numbers is divisible by 9? (a) 7014 (b) 7021 (c) 7020 (d) 7030
11. 11.What is the smallest number you must add to 293 to make it divisible by 12? (a) 7 (b) 5 (c) 6 (d) 8
12. 12.What is the smallest number you must add to 211 to make it divisible by 4? (a) 1 (b) 3 (c) 2
13. 13.Which of these numbers is divisible by 6? (a) 3310 (b) 3298 (c) 3301 (d) 3300
14. 14.What is the smallest number you must add to 297 to make it divisible by 4? (a) 1 (b) 3 (c) 2
15. 15.What is the units digit of 7^23? (a) 7 (b) 1 (c) 3 (d) 9
16. 16.What is the smallest number you must add to 376 to make it divisible by 10? (a) 3 (b) 5 (c) 4 (d) 6
17. 17.What is the smallest number you must add to 822 to make it divisible by 11? (a) 4 (b) 2 (c) 3 (d) 8
18. 18.What is the units digit of 2^52? (a) 4 (b) 2 (c) 6 (d) 8
19. 19.Which of these numbers is divisible by 11? (a) 4029 (b) 4026 (c) 4025 (d) 4041
20. 20.What is the units digit of 2^51? (a) 6 (b) 8 (c) 4 (d) 2
21. 21.Which of these numbers is divisible by 3? (a) 313 (b) 310 (c) 317 (d) 312
22. 22.What is the units digit of 8^55? (a) 4 (b) 6 (c) 2 (d) 8
23. 23.Which of these numbers is divisible by 4? (a) 245 (b) 241 (c) 251 (d) 244
24. 24.What is the remainder when 94 is divided by 4? (a) 3 (b) 2 (c) 0 (d) 1
25. 25.Which of these numbers is divisible by 11? (a) 5131 (b) 5126 (c) 5119 (d) 5143
26. 26.What is the units digit of 4^24? (a) 7 (b) 6 (c) 5 (d) 4
27. 27.Which of these numbers is divisible by 3? (a) 2567 (b) 2568 (c) 2569 (d) 2572
28. 28.Which of these numbers is divisible by 3? (a) 1433 (b) 1430 (c) 1426 (d) 1428
29. 29.Which of these numbers is divisible by 4? (a) 1820 (b) 1823 (c) 1827 (d) 1818
30. 30.What is the remainder when 328 is divided by 11? (a) 8 (b) 9 (c) 10 (d) 2
31. 31.What is the remainder when 242 is divided by 3? (a) 0 (b) 2 (c) 1
32. 32.What is the smallest number you must add to 260 to make it divisible by 11? (a) 4 (b) 5 (c) 3 (d) 7
33. 33.What is the units digit of 4^14? (a) 6 (b) 4 (c) 5 (d) 7
34. 34.Which of these numbers is divisible by 6? (a) 1455 (b) 1441 (c) 1446 (d) 1447
35. 35.Which of these numbers is divisible by 4? (a) 2709 (b) 2719 (c) 2713 (d) 2712
36. 36.Which of these numbers is divisible by 11? (a) 8162 (b) 8168 (c) 8159 (d) 8181
37. 37.Which of these numbers is divisible by 6? (a) 4198 (b) 4188 (c) 4185 (d) 4191
38. 38.What is the units digit of 9^24? (a) 2 (b) 0 (c) 9 (d) 1
39. 39.What is the smallest number you must add to 290 to make it divisible by 4? (a) 2 (b) 3 (c) 1
40. 40.What is the remainder when 263 is divided by 5? (a) 1 (b) 4 (c) 2 (d) 3

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