Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 142 · 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 76 is divided by 5? (a) 4 (b) 1 (c) 0 (d) 2
2. 2.Which of these numbers is divisible by 11? (a) 4084 (b) 4070 (c) 4075 (d) 4069
3. 3.Which of these numbers is divisible by 3? (a) 1377 (b) 1381 (c) 1379 (d) 1376
4. 4.What is the units digit of 7^42? (a) 1 (b) 7 (c) 9 (d) 3
5. 5.What is the remainder when 285 is divided by 8? (a) 5 (b) 3 (c) 4 (d) 6
6. 6.What is the remainder when 663 is divided by 9? (a) 6 (b) 7 (c) 3 (d) 5
7. 7.What is the units digit of 9^59? (a) 8 (b) 1 (c) 9 (d) 7
8. 8.Which of these numbers is divisible by 8? (a) 2505 (b) 2512 (c) 2513 (d) 2526
9. 9.What is the units digit of 2^45? (a) 4 (b) 6 (c) 2 (d) 8
10. 10.What is the smallest number you must add to 237 to make it divisible by 4? (a) 3 (b) 1 (c) 2
11. 11.What is the units digit of 9^31? (a) 8 (b) 7 (c) 9 (d) 1
12. 12.Which of these numbers is divisible by 8? (a) 1954 (b) 1947 (c) 1967 (d) 1952
13. 13.What is the remainder when 322 is divided by 8? (a) 2 (b) 6 (c) 1 (d) 3
14. 14.What is the remainder when 122 is divided by 8? (a) 6 (b) 3 (c) 1 (d) 2
15. 15.Which of these numbers is divisible by 8? (a) 3378 (b) 3365 (c) 3368 (d) 3369
16. 16.Which of these numbers is divisible by 11? (a) 8743 (b) 8734 (c) 8750 (d) 8728
17. 17.Which of these numbers is divisible by 3? (a) 2183 (b) 2185 (c) 2184 (d) 2189
18. 18.What is the smallest number you must add to 145 to make it divisible by 8? (a) 1 (b) 6 (c) 5 (d) 7
19. 19.Which of these numbers is divisible by 11? (a) 6292 (b) 6295 (c) 6307 (d) 6285
20. 20.What is the smallest number you must add to 106 to make it divisible by 4? (a) 3 (b) 2 (c) 1
21. 21.Which of these numbers is divisible by 9? (a) 473 (b) 481 (c) 460 (d) 468
22. 22.What is the smallest number you must add to 337 to make it divisible by 9? (a) 4 (b) 7 (c) 6 (d) 5
23. 23.What is the smallest number you must add to 356 to make it divisible by 7? (a) 3 (b) 2 (c) 1 (d) 6
24. 24.Which of these numbers is divisible by 6? (a) 2297 (b) 2298 (c) 2305 (d) 2299
25. 25.What is the smallest number you must add to 409 to make it divisible by 12? (a) 1 (b) 10 (c) 9 (d) 11
26. 26.What is the remainder when 82 is divided by 3? (a) 2 (b) 0 (c) 1
27. 27.What is the units digit of 4^26? (a) 5 (b) 4 (c) 6 (d) 7
28. 28.What is the smallest number you must add to 202 to make it divisible by 11? (a) 4 (b) 8 (c) 7 (d) 6
29. 29.What is the remainder when 437 is divided by 6? (a) 1 (b) 3 (c) 5 (d) 4
30. 30.What is the remainder when 483 is divided by 9? (a) 3 (b) 7 (c) 5 (d) 6
31. 31.What is the smallest number you must add to 171 to make it divisible by 11? (a) 4 (b) 7 (c) 5 (d) 6
32. 32.What is the remainder when 626 is divided by 10? (a) 4 (b) 5 (c) 6 (d) 7
33. 33.Which of these numbers is divisible by 8? (a) 3348 (b) 3337 (c) 3344 (d) 3358
34. 34.What is the remainder when 608 is divided by 10? (a) 2 (b) 7 (c) 8 (d) 9
35. 35.What is the remainder when 388 is divided by 5? (a) 1 (b) 2 (c) 3 (d) 4
36. 36.What is the remainder when 131 is divided by 8? (a) 4 (b) 2 (c) 3 (d) 5
37. 37.Which of these numbers is divisible by 6? (a) 2560 (b) 2562 (c) 2565 (d) 2572
38. 38.Which of these numbers is divisible by 9? (a) 2452 (b) 2446 (c) 2448 (d) 2465
39. 39.What is the smallest number you must add to 234 to make it divisible by 8? (a) 2 (b) 7 (c) 6 (d) 5
40. 40.What is the smallest number you must add to 218 to make it divisible by 7? (a) 1 (b) 6 (c) 5 (d) 4

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