Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 42 · 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) 5518 (b) 5512 (c) 5511 (d) 5521
2. 2.What is the remainder when 265 is divided by 4? (a) 1 (b) 3 (c) 2 (d) 0
3. 3.Which of these numbers is divisible by 3? (a) 315 (b) 319 (c) 314 (d) 316
4. 4.What is the units digit of 2^27? (a) 8 (b) 2 (c) 4 (d) 6
5. 5.What is the smallest number you must add to 76 to make it divisible by 5? (a) 3 (b) 1 (c) 4 (d) 2
6. 6.What is the remainder when 79 is divided by 5? (a) 3 (b) 1 (c) 4 (d) 2
7. 7.What is the units digit of 7^22? (a) 7 (b) 3 (c) 1 (d) 9
8. 8.What is the remainder when 86 is divided by 5? (a) 0 (b) 1 (c) 2 (d) 4
9. 9.What is the smallest number you must add to 538 to make it divisible by 7? (a) 1 (b) 2 (c) 6 (d) 3
10. 10.What is the smallest number you must add to 391 to make it divisible by 9? (a) 7 (b) 5 (c) 6 (d) 4
11. 11.What is the smallest number you must add to 424 to make it divisible by 11? (a) 6 (b) 7 (c) 5 (d) 4
12. 12.What is the remainder when 461 is divided by 8? (a) 3 (b) 6 (c) 4 (d) 5
13. 13.What is the units digit of 3^11? (a) 9 (b) 3 (c) 7 (d) 1
14. 14.What is the smallest number you must add to 226 to make it divisible by 12? (a) 2 (b) 3 (c) 1 (d) 10
15. 15.What is the remainder when 506 is divided by 7? (a) 3 (b) 5 (c) 1 (d) 2
16. 16.Which of these numbers is divisible by 9? (a) 3401 (b) 3386 (c) 3393 (d) 3408
17. 17.Which of these numbers is divisible by 9? (a) 7200 (b) 7214 (c) 7204 (d) 7199
18. 18.What is the smallest number you must add to 772 to make it divisible by 10? (a) 7 (b) 2 (c) 8 (d) 9
19. 19.Which of these numbers is divisible by 11? (a) 1190 (b) 1177 (c) 1178 (d) 1169
20. 20.What is the smallest number you must add to 689 to make it divisible by 11? (a) 4 (b) 3 (c) 5 (d) 7
21. 21.What is the remainder when 904 is divided by 12? (a) 4 (b) 3 (c) 8 (d) 5
22. 22.What is the units digit of 9^60? (a) 2 (b) 0 (c) 9 (d) 1
23. 23.What is the remainder when 508 is divided by 11? (a) 9 (b) 3 (c) 1 (d) 2
24. 24.What is the remainder when 74 is divided by 3? (a) 0 (b) 1 (c) 2
25. 25.What is the remainder when 164 is divided by 9? (a) 3 (b) 2 (c) 7 (d) 1
26. 26.What is the smallest number you must add to 211 to make it divisible by 8? (a) 5 (b) 3 (c) 6 (d) 4
27. 27.What is the remainder when 156 is divided by 9? (a) 3 (b) 4 (c) 6 (d) 2
28. 28.Which of these numbers is divisible by 4? (a) 2635 (b) 2629 (c) 2637 (d) 2632
29. 29.What is the units digit of 2^12? (a) 6 (b) 8 (c) 2 (d) 4
30. 30.Which of these numbers is divisible by 3? (a) 1249 (b) 1247 (c) 1248 (d) 1253
31. 31.Which of these numbers is divisible by 9? (a) 462 (b) 446 (c) 451 (d) 450
32. 32.Which of these numbers is divisible by 4? (a) 3338 (b) 3343 (c) 3336 (d) 3335
33. 33.Which of these numbers is divisible by 4? (a) 3241 (b) 3246 (c) 3251 (d) 3244
34. 34.What is the smallest number you must add to 101 to make it divisible by 6? (a) 3 (b) 1 (c) 2 (d) 5
35. 35.What is the units digit of 4^37? (a) 5 (b) 6 (c) 4 (d) 3
36. 36.What is the smallest number you must add to 250 to make it divisible by 6? (a) 2 (b) 3 (c) 4 (d) 1
37. 37.Which of these numbers is divisible by 11? (a) 447 (b) 451 (c) 459 (d) 463
38. 38.What is the units digit of 4^18? (a) 5 (b) 4 (c) 6 (d) 7
39. 39.Which of these numbers is divisible by 4? (a) 672 (b) 678 (c) 671 (d) 675
40. 40.What is the units digit of 4^40? (a) 4 (b) 5 (c) 6 (d) 7

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