Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 605 · 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 663 is divided by 11? (a) 4 (b) 3 (c) 8 (d) 2
2. 2.Which of these numbers is divisible by 9? (a) 5283 (b) 5290 (c) 5282 (d) 5300
3. 3.What is the remainder when 378 is divided by 8? (a) 1 (b) 2 (c) 6 (d) 3
4. 4.Which of these numbers is divisible by 6? (a) 660 (b) 656 (c) 661 (d) 667
5. 5.What is the remainder when 408 is divided by 7? (a) 1 (b) 3 (c) 2 (d) 5
6. 6.Which of these numbers is divisible by 6? (a) 3821 (b) 3827 (c) 3822 (d) 3833
7. 7.Which of these numbers is divisible by 11? (a) 8395 (b) 8416 (c) 8412 (d) 8404
8. 8.What is the remainder when 143 is divided by 12? (a) 10 (b) 11 (c) 1 (d) 9
9. 9.What is the units digit of 7^51? (a) 3 (b) 9 (c) 1 (d) 7
10. 10.What is the remainder when 333 is divided by 6? (a) 5 (b) 4 (c) 2 (d) 3
11. 11.What is the remainder when 101 is divided by 8? (a) 4 (b) 3 (c) 5 (d) 6
12. 12.What is the units digit of 2^40? (a) 4 (b) 6 (c) 8 (d) 2
13. 13.Which of these numbers is divisible by 9? (a) 390 (b) 402 (c) 380 (d) 387
14. 14.What is the remainder when 89 is divided by 7? (a) 2 (b) 5 (c) 6 (d) 4
15. 15.What is the smallest number you must add to 161 to make it divisible by 4? (a) 3 (b) 2 (c) 1
16. 16.What is the units digit of 4^53? (a) 4 (b) 5 (c) 6 (d) 3
17. 17.What is the smallest number you must add to 254 to make it divisible by 10? (a) 7 (b) 4 (c) 5 (d) 6
18. 18.What is the remainder when 914 is divided by 12? (a) 1 (b) 3 (c) 2 (d) 10
19. 19.What is the smallest number you must add to 823 to make it divisible by 12? (a) 7 (b) 4 (c) 6 (d) 5
20. 20.Which of these numbers is divisible by 3? (a) 379 (b) 375 (c) 373 (d) 376
21. 21.Which of these numbers is divisible by 11? (a) 3601 (b) 3576 (c) 3594 (d) 3586
22. 22.What is the remainder when 237 is divided by 8? (a) 3 (b) 6 (c) 4 (d) 5
23. 23.What is the remainder when 332 is divided by 5? (a) 1 (b) 2 (c) 4 (d) 3
24. 24.What is the remainder when 95 is divided by 4? (a) 2 (b) 1 (c) 0 (d) 3
25. 25.What is the smallest number you must add to 268 to make it divisible by 7? (a) 2 (b) 4 (c) 6 (d) 5
26. 26.What is the remainder when 212 is divided by 7? (a) 3 (b) 2 (c) 1 (d) 5
27. 27.What is the remainder when 135 is divided by 4? (a) 0 (b) 2 (c) 3 (d) 1
28. 28.Which of these numbers is divisible by 8? (a) 2192 (b) 2202 (c) 2199 (d) 2189
29. 29.Which of these numbers is divisible by 4? (a) 490 (b) 494 (c) 487 (d) 488
30. 30.Which of these numbers is divisible by 9? (a) 3075 (b) 3078 (c) 3094 (d) 3083
31. 31.Which of these numbers is divisible by 11? (a) 1584 (b) 1596 (c) 1580 (d) 1586
32. 32.What is the remainder when 232 is divided by 3? (a) 2 (b) 0 (c) 1
33. 33.What is the smallest number you must add to 224 to make it divisible by 3? (a) 2 (b) 1
34. 34.What is the smallest number you must add to 867 to make it divisible by 11? (a) 3 (b) 9 (c) 1 (d) 2
35. 35.What is the units digit of 9^31? (a) 7 (b) 1 (c) 8 (d) 9
36. 36.What is the remainder when 91 is divided by 8? (a) 5 (b) 2 (c) 4 (d) 3
37. 37.Which of these numbers is divisible by 3? (a) 1484 (b) 1481 (c) 1482 (d) 1486
38. 38.What is the units digit of 2^19? (a) 4 (b) 6 (c) 8 (d) 2
39. 39.Which of these numbers is divisible by 3? (a) 2338 (b) 2337 (c) 2341 (d) 2336
40. 40.Which of these numbers is divisible by 9? (a) 1614 (b) 1607 (c) 1611 (d) 1625

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