Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 948 · 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 smallest number you must add to 289 to make it divisible by 8? (a) 1 (b) 7 (c) 5 (d) 6
2. 2.What is the remainder when 124 is divided by 7? (a) 4 (b) 6 (c) 5 (d) 2
3. 3.Which of these numbers is divisible by 6? (a) 1223 (b) 1234 (c) 1226 (d) 1224
4. 4.What is the remainder when 399 is divided by 11? (a) 4 (b) 8 (c) 3 (d) 2
5. 5.Which of these numbers is divisible by 6? (a) 2013 (b) 2026 (c) 2017 (d) 2016
6. 6.Which of these numbers is divisible by 8? (a) 1124 (b) 1112 (c) 1106 (d) 1115
7. 7.What is the smallest number you must add to 339 to make it divisible by 7? (a) 4 (b) 3 (c) 5 (d) 6
8. 8.What is the units digit of 2^37? (a) 2 (b) 4 (c) 6 (d) 8
9. 9.What is the units digit of 8^28? (a) 8 (b) 4 (c) 2 (d) 6
10. 10.What is the units digit of 9^18? (a) 0 (b) 9 (c) 2 (d) 1
11. 11.What is the remainder when 146 is divided by 11? (a) 4 (b) 3 (c) 2 (d) 8
12. 12.Which of these numbers is divisible by 8? (a) 422 (b) 439 (c) 424 (d) 425
13. 13.What is the smallest number you must add to 285 to make it divisible by 4? (a) 2 (b) 3 (c) 1
14. 14.What is the smallest number you must add to 225 to make it divisible by 7? (a) 6 (b) 4 (c) 1 (d) 5
15. 15.What is the units digit of 3^23? (a) 9 (b) 1 (c) 7 (d) 3
16. 16.What is the units digit of 9^59? (a) 7 (b) 1 (c) 8 (d) 9
17. 17.What is the remainder when 651 is divided by 11? (a) 3 (b) 1 (c) 9 (d) 2
18. 18.What is the smallest number you must add to 86 to make it divisible by 3? (a) 2 (b) 1
19. 19.What is the remainder when 306 is divided by 12? (a) 8 (b) 5 (c) 6 (d) 7
20. 20.What is the remainder when 282 is divided by 9? (a) 6 (b) 4 (c) 2 (d) 3
21. 21.What is the remainder when 611 is divided by 12? (a) 9 (b) 1 (c) 10 (d) 11
22. 22.What is the remainder when 474 is divided by 7? (a) 5 (b) 6 (c) 2 (d) 4
23. 23.What is the remainder when 349 is divided by 6? (a) 0 (b) 5 (c) 1 (d) 2
24. 24.What is the units digit of 7^17? (a) 7 (b) 3 (c) 1 (d) 9
25. 25.What is the remainder when 110 is divided by 4? (a) 1 (b) 0 (c) 2 (d) 3
26. 26.Which of these numbers is divisible by 4? (a) 2654 (b) 2658 (c) 2652 (d) 2649
27. 27.Which of these numbers is divisible by 3? (a) 1466 (b) 1472 (c) 1467 (d) 1468
28. 28.Which of these numbers is divisible by 6? (a) 1023 (b) 1029 (c) 1020 (d) 1018
29. 29.What is the smallest number you must add to 177 to make it divisible by 4? (a) 1 (b) 2 (c) 3
30. 30.What is the smallest number you must add to 358 to make it divisible by 5? (a) 3 (b) 1 (c) 4 (d) 2
31. 31.What is the units digit of 7^60? (a) 9 (b) 1 (c) 3 (d) 7
32. 32.What is the units digit of 7^12? (a) 9 (b) 3 (c) 1 (d) 7
33. 33.What is the smallest number you must add to 55 to make it divisible by 4? (a) 2 (b) 3 (c) 1
34. 34.What is the smallest number you must add to 317 to make it divisible by 10? (a) 2 (b) 4 (c) 7 (d) 3
35. 35.What is the remainder when 145 is divided by 3? (a) 2 (b) 0 (c) 1
36. 36.What is the remainder when 195 is divided by 9? (a) 7 (b) 5 (c) 6 (d) 3
37. 37.What is the units digit of 2^47? (a) 6 (b) 2 (c) 8 (d) 4
38. 38.Which of these numbers is divisible by 9? (a) 3576 (b) 3584 (c) 3582 (d) 3593
39. 39.Which of these numbers is divisible by 9? (a) 1317 (b) 1307 (c) 1305 (d) 1297
40. 40.What is the smallest number you must add to 247 to make it divisible by 4? (a) 2 (b) 3 (c) 1

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