Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 637 · 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 223 to make it divisible by 8? (a) 1 (b) 2 (c) 3 (d) 7
2. 2.What is the smallest number you must add to 117 to make it divisible by 6? (a) 5 (b) 2 (c) 4 (d) 3
3. 3.What is the remainder when 46 is divided by 4? (a) 3 (b) 0 (c) 1 (d) 2
4. 4.What is the remainder when 209 is divided by 6? (a) 1 (b) 3 (c) 5 (d) 4
5. 5.What is the remainder when 547 is divided by 8? (a) 5 (b) 2 (c) 4 (d) 3
6. 6.What is the units digit of 2^11? (a) 8 (b) 6 (c) 4 (d) 2
7. 7.What is the remainder when 183 is divided by 4? (a) 0 (b) 2 (c) 1 (d) 3
8. 8.What is the units digit of 7^39? (a) 9 (b) 1 (c) 3 (d) 7
9. 9.What is the remainder when 223 is divided by 11? (a) 3 (b) 2 (c) 8 (d) 4
10. 10.What is the remainder when 507 is divided by 8? (a) 3 (b) 5 (c) 4 (d) 2
11. 11.Which of these numbers is divisible by 8? (a) 330 (b) 313 (c) 320 (d) 327
12. 12.Which of these numbers is divisible by 8? (a) 558 (b) 544 (c) 539 (d) 548
13. 13.What is the units digit of 8^19? (a) 6 (b) 2 (c) 8 (d) 4
14. 14.What is the smallest number you must add to 698 to make it divisible by 9? (a) 6 (b) 5 (c) 4 (d) 3
15. 15.What is the smallest number you must add to 273 to make it divisible by 5? (a) 3 (b) 2 (c) 4 (d) 1
16. 16.What is the smallest number you must add to 539 to make it divisible by 10? (a) 9 (b) 3 (c) 1 (d) 2
17. 17.What is the units digit of 4^22? (a) 7 (b) 4 (c) 5 (d) 6
18. 18.Which of these numbers is divisible by 3? (a) 482 (b) 475 (c) 477 (d) 478
19. 19.Which of these numbers is divisible by 9? (a) 702 (b) 713 (c) 704 (d) 699
20. 20.What is the smallest number you must add to 116 to make it divisible by 5? (a) 1 (b) 3 (c) 2 (d) 4
21. 21.Which of these numbers is divisible by 11? (a) 6193 (b) 6214 (c) 6200 (d) 6187
22. 22.What is the units digit of 7^57? (a) 3 (b) 7 (c) 9 (d) 1
23. 23.What is the smallest number you must add to 168 to make it divisible by 5? (a) 3 (b) 4 (c) 2 (d) 1
24. 24.What is the units digit of 3^12? (a) 3 (b) 7 (c) 9 (d) 1
25. 25.Which of these numbers is divisible by 4? (a) 2769 (b) 2763 (c) 2764 (d) 2766
26. 26.Which of these numbers is divisible by 3? (a) 188 (b) 184 (c) 183 (d) 182
27. 27.Which of these numbers is divisible by 9? (a) 3358 (b) 3353 (c) 3372 (d) 3357
28. 28.What is the remainder when 676 is divided by 10? (a) 5 (b) 6 (c) 7 (d) 4
29. 29.What is the units digit of 2^13? (a) 6 (b) 8 (c) 4 (d) 2
30. 30.What is the remainder when 68 is divided by 3? (a) 2 (b) 0 (c) 1
31. 31.What is the units digit of 2^33? (a) 2 (b) 8 (c) 4 (d) 6
32. 32.What is the smallest number you must add to 110 to make it divisible by 8? (a) 1 (b) 6 (c) 2 (d) 3
33. 33.What is the remainder when 142 is divided by 12? (a) 10 (b) 9 (c) 2 (d) 11
34. 34.Which of these numbers is divisible by 6? (a) 4207 (b) 4219 (c) 4212 (d) 4214
35. 35.Which of these numbers is divisible by 4? (a) 698 (b) 692 (c) 691 (d) 694
36. 36.What is the units digit of 9^39? (a) 7 (b) 8 (c) 1 (d) 9
37. 37.What is the remainder when 722 is divided by 10? (a) 3 (b) 2 (c) 1 (d) 8
38. 38.Which of these numbers is divisible by 9? (a) 6529 (b) 6518 (c) 6508 (d) 6516
39. 39.Which of these numbers is divisible by 9? (a) 1711 (b) 1707 (c) 1720 (d) 1710
40. 40.What is the remainder when 233 is divided by 5? (a) 2 (b) 1 (c) 4 (d) 3

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