Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 380 · 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.Which of these numbers is divisible by 8? (a) 3939 (b) 3933 (c) 3949 (d) 3936
2. 2.What is the smallest number you must add to 261 to make it divisible by 8? (a) 5 (b) 4 (c) 2 (d) 3
3. 3.What is the remainder when 82 is divided by 8? (a) 3 (b) 2 (c) 1 (d) 6
4. 4.Which of these numbers is divisible by 3? (a) 479 (b) 485 (c) 480 (d) 482
5. 5.What is the units digit of 2^14? (a) 8 (b) 6 (c) 2 (d) 4
6. 6.What is the remainder when 890 is divided by 12? (a) 3 (b) 2 (c) 1 (d) 10
7. 7.Which of these numbers is divisible by 9? (a) 4643 (b) 4631 (c) 4626 (d) 4619
8. 8.Which of these numbers is divisible by 9? (a) 4089 (b) 4095 (c) 4108 (d) 4096
9. 9.What is the smallest number you must add to 281 to make it divisible by 4? (a) 2 (b) 1 (c) 3
10. 10.What is the units digit of 3^35? (a) 3 (b) 7 (c) 1 (d) 9
11. 11.What is the remainder when 46 is divided by 4? (a) 2 (b) 3 (c) 0 (d) 1
12. 12.Which of these numbers is divisible by 9? (a) 3471 (b) 3456 (c) 3451 (d) 3459
13. 13.What is the smallest number you must add to 290 to make it divisible by 6? (a) 3 (b) 4 (c) 5 (d) 2
14. 14.What is the units digit of 7^18? (a) 7 (b) 1 (c) 3 (d) 9
15. 15.Which of these numbers is divisible by 6? (a) 4522 (b) 4532 (c) 4524 (d) 4527
16. 16.Which of these numbers is divisible by 6? (a) 2671 (b) 2678 (c) 2676 (d) 2686
17. 17.Which of these numbers is divisible by 4? (a) 1719 (b) 1712 (c) 1714 (d) 1711
18. 18.Which of these numbers is divisible by 4? (a) 1159 (b) 1162 (c) 1160 (d) 1167
19. 19.What is the remainder when 397 is divided by 12? (a) 2 (b) 0 (c) 11 (d) 1
20. 20.Which of these numbers is divisible by 6? (a) 327 (b) 321 (c) 313 (d) 318
21. 21.What is the smallest number you must add to 212 to make it divisible by 11? (a) 3 (b) 9 (c) 8 (d) 7
22. 22.What is the units digit of 2^47? (a) 4 (b) 2 (c) 8 (d) 6
23. 23.What is the units digit of 3^27? (a) 9 (b) 7 (c) 3 (d) 1
24. 24.What is the smallest number you must add to 392 to make it divisible by 11? (a) 7 (b) 5 (c) 3 (d) 4
25. 25.Which of these numbers is divisible by 4? (a) 1074 (b) 1067 (c) 1068 (d) 1071
26. 26.Which of these numbers is divisible by 8? (a) 3311 (b) 3295 (c) 3299 (d) 3296
27. 27.What is the smallest number you must add to 584 to make it divisible by 9? (a) 8 (b) 3 (c) 1 (d) 2
28. 28.What is the smallest number you must add to 342 to make it divisible by 11? (a) 1 (b) 9 (c) 8 (d) 10
29. 29.What is the smallest number you must add to 305 to make it divisible by 8? (a) 5 (b) 6 (c) 7 (d) 1
30. 30.What is the units digit of 2^26? (a) 8 (b) 2 (c) 4 (d) 6
31. 31.What is the smallest number you must add to 277 to make it divisible by 12? (a) 1 (b) 9 (c) 10 (d) 11
32. 32.What is the remainder when 197 is divided by 3? (a) 2 (b) 1 (c) 0
33. 33.Which of these numbers is divisible by 11? (a) 6704 (b) 6688 (c) 6687 (d) 6696
34. 34.What is the remainder when 389 is divided by 10? (a) 9 (b) 8 (c) 7 (d) 1
35. 35.Which of these numbers is divisible by 3? (a) 2228 (b) 2224 (c) 2221 (d) 2223
36. 36.What is the units digit of 2^22? (a) 8 (b) 4 (c) 2 (d) 6
37. 37.What is the remainder when 559 is divided by 7? (a) 1 (b) 4 (c) 6 (d) 5
38. 38.Which of these numbers is divisible by 6? (a) 4846 (b) 4848 (c) 4851 (d) 4859
39. 39.What is the smallest number you must add to 40 to make it divisible by 3? (a) 2 (b) 1
40. 40.What is the units digit of 3^37? (a) 7 (b) 9 (c) 1 (d) 3

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