Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 625 · 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 153 to make it divisible by 4? (a) 2 (b) 1 (c) 3
2. 2.Which of these numbers is divisible by 3? (a) 539 (b) 536 (c) 541 (d) 537
3. 3.What is the remainder when 701 is divided by 11? (a) 3 (b) 9 (c) 8 (d) 7
4. 4.What is the remainder when 362 is divided by 6? (a) 3 (b) 2 (c) 1 (d) 4
5. 5.Which of these numbers is divisible by 11? (a) 5368 (b) 5363 (c) 5369 (d) 5388
6. 6.Which of these numbers is divisible by 11? (a) 2322 (b) 2311 (c) 2339 (d) 2321
7. 7.What is the remainder when 417 is divided by 8? (a) 7 (b) 2 (c) 1 (d) 0
8. 8.What is the remainder when 198 is divided by 7? (a) 3 (b) 5 (c) 2 (d) 1
9. 9.What is the units digit of 7^28? (a) 7 (b) 3 (c) 9 (d) 1
10. 10.What is the remainder when 227 is divided by 4? (a) 2 (b) 1 (c) 0 (d) 3
11. 11.What is the units digit of 7^48? (a) 1 (b) 7 (c) 3 (d) 9
12. 12.What is the smallest number you must add to 296 to make it divisible by 5? (a) 3 (b) 1 (c) 2 (d) 4
13. 13.What is the smallest number you must add to 410 to make it divisible by 6? (a) 3 (b) 2 (c) 5 (d) 4
14. 14.What is the remainder when 626 is divided by 12? (a) 1 (b) 2 (c) 3 (d) 10
15. 15.Which of these numbers is divisible by 11? (a) 8699 (b) 8721 (c) 8701 (d) 8703
16. 16.What is the remainder when 404 is divided by 5? (a) 4 (b) 1 (c) 2 (d) 3
17. 17.What is the units digit of 7^16? (a) 3 (b) 1 (c) 7 (d) 9
18. 18.What is the units digit of 8^31? (a) 8 (b) 2 (c) 6 (d) 4
19. 19.Which of these numbers is divisible by 9? (a) 8113 (b) 8100 (c) 8097 (d) 8107
20. 20.Which of these numbers is divisible by 11? (a) 5524 (b) 5511 (c) 5507 (d) 5513
21. 21.What is the units digit of 9^39? (a) 1 (b) 9 (c) 7 (d) 8
22. 22.What is the smallest number you must add to 184 to make it divisible by 7? (a) 4 (b) 6 (c) 5 (d) 2
23. 23.What is the smallest number you must add to 313 to make it divisible by 5? (a) 3 (b) 1 (c) 2 (d) 4
24. 24.What is the remainder when 73 is divided by 4? (a) 3 (b) 0 (c) 2 (d) 1
25. 25.What is the remainder when 341 is divided by 6? (a) 3 (b) 5 (c) 4 (d) 1
26. 26.What is the remainder when 88 is divided by 3? (a) 0 (b) 2 (c) 1
27. 27.Which of these numbers is divisible by 8? (a) 420 (b) 430 (c) 412 (d) 416
28. 28.Which of these numbers is divisible by 8? (a) 5193 (b) 5192 (c) 5185 (d) 5202
29. 29.What is the smallest number you must add to 169 to make it divisible by 8? (a) 7 (b) 5 (c) 6 (d) 1
30. 30.Which of these numbers is divisible by 3? (a) 740 (b) 737 (c) 738 (d) 743
31. 31.What is the smallest number you must add to 299 to make it divisible by 9? (a) 2 (b) 6 (c) 7 (d) 8
32. 32.What is the smallest number you must add to 709 to make it divisible by 10? (a) 3 (b) 2 (c) 9 (d) 1
33. 33.What is the smallest number you must add to 181 to make it divisible by 10? (a) 9 (b) 8 (c) 7 (d) 1
34. 34.What is the smallest number you must add to 464 to make it divisible by 9? (a) 6 (b) 4 (c) 5 (d) 3
35. 35.Which of these numbers is divisible by 4? (a) 1548 (b) 1547 (c) 1551 (d) 1555
36. 36.What is the remainder when 67 is divided by 6? (a) 2 (b) 0 (c) 5 (d) 1
37. 37.What is the remainder when 208 is divided by 6? (a) 2 (b) 5 (c) 4 (d) 3
38. 38.Which of these numbers is divisible by 4? (a) 2587 (b) 2595 (c) 2589 (d) 2588
39. 39.Which of these numbers is divisible by 6? (a) 2392 (b) 2385 (c) 2388 (d) 2397
40. 40.What is the smallest number you must add to 290 to make it divisible by 4? (a) 1 (b) 2 (c) 3

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