Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 390 · 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 327 to make it divisible by 6? (a) 4 (b) 5 (c) 2 (d) 3
2. 2.What is the smallest number you must add to 155 to make it divisible by 9? (a) 7 (b) 2 (c) 8 (d) 6
3. 3.What is the remainder when 221 is divided by 5? (a) 2 (b) 4 (c) 0 (d) 1
4. 4.What is the smallest number you must add to 80 to make it divisible by 3? (a) 2 (b) 1
5. 5.Which of these numbers is divisible by 4? (a) 735 (b) 742 (c) 738 (d) 736
6. 6.What is the units digit of 2^29? (a) 8 (b) 4 (c) 2 (d) 6
7. 7.What is the remainder when 848 is divided by 11? (a) 10 (b) 0 (c) 1 (d) 2
8. 8.What is the smallest number you must add to 563 to make it divisible by 9? (a) 4 (b) 3 (c) 6 (d) 5
9. 9.What is the units digit of 9^21? (a) 9 (b) 1 (c) 8 (d) 7
10. 10.What is the smallest number you must add to 307 to make it divisible by 12? (a) 5 (b) 4 (c) 6 (d) 7
11. 11.What is the units digit of 2^50? (a) 6 (b) 8 (c) 2 (d) 4
12. 12.What is the units digit of 8^35? (a) 6 (b) 2 (c) 8 (d) 4
13. 13.What is the units digit of 9^47? (a) 8 (b) 7 (c) 9 (d) 1
14. 14.What is the units digit of 7^11? (a) 1 (b) 9 (c) 3 (d) 7
15. 15.Which of these numbers is divisible by 4? (a) 351 (b) 355 (c) 352 (d) 358
16. 16.Which of these numbers is divisible by 4? (a) 308 (b) 309 (c) 306 (d) 314
17. 17.Which of these numbers is divisible by 9? (a) 3308 (b) 3303 (c) 3318 (d) 3295
18. 18.Which of these numbers is divisible by 3? (a) 1801 (b) 1808 (c) 1805 (d) 1803
19. 19.What is the remainder when 304 is divided by 10? (a) 5 (b) 3 (c) 6 (d) 4
20. 20.What is the smallest number you must add to 335 to make it divisible by 11? (a) 8 (b) 5 (c) 6 (d) 7
21. 21.What is the smallest number you must add to 613 to make it divisible by 12? (a) 9 (b) 10 (c) 1 (d) 11
22. 22.What is the remainder when 232 is divided by 5? (a) 3 (b) 1 (c) 4 (d) 2
23. 23.Which of these numbers is divisible by 8? (a) 6838 (b) 6825 (c) 6819 (d) 6824
24. 24.Which of these numbers is divisible by 6? (a) 983 (b) 991 (c) 984 (d) 985
25. 25.Which of these numbers is divisible by 8? (a) 2536 (b) 2534 (c) 2542 (d) 2545
26. 26.What is the remainder when 552 is divided by 10? (a) 1 (b) 3 (c) 2 (d) 8
27. 27.What is the units digit of 3^49? (a) 3 (b) 9 (c) 1 (d) 7
28. 28.What is the remainder when 176 is divided by 5? (a) 2 (b) 0 (c) 4 (d) 1
29. 29.What is the remainder when 532 is divided by 10? (a) 8 (b) 3 (c) 1 (d) 2
30. 30.Which of these numbers is divisible by 8? (a) 6285 (b) 6272 (c) 6276 (d) 6267
31. 31.Which of these numbers is divisible by 9? (a) 2043 (b) 2036 (c) 2056 (d) 2045
32. 32.What is the remainder when 89 is divided by 8? (a) 1 (b) 2 (c) 7 (d) 0
33. 33.What is the smallest number you must add to 835 to make it divisible by 11? (a) 2 (b) 1 (c) 10 (d) 3
34. 34.What is the remainder when 118 is divided by 6? (a) 3 (b) 4 (c) 2 (d) 5
35. 35.What is the remainder when 696 is divided by 10? (a) 5 (b) 4 (c) 6 (d) 7
36. 36.What is the units digit of 9^11? (a) 1 (b) 8 (c) 9 (d) 7
37. 37.What is the remainder when 378 is divided by 5? (a) 1 (b) 3 (c) 2 (d) 4
38. 38.What is the remainder when 143 is divided by 5? (a) 4 (b) 3 (c) 2 (d) 1
39. 39.Which of these numbers is divisible by 6? (a) 1750 (b) 1756 (c) 1752 (d) 1759
40. 40.What is the remainder when 215 is divided by 6? (a) 4 (b) 3 (c) 5 (d) 1

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