Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 639 · 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 320 is divided by 11? (a) 1 (b) 2 (c) 10 (d) 0
2. 2.What is the remainder when 378 is divided by 5? (a) 2 (b) 1 (c) 4 (d) 3
3. 3.Which of these numbers is divisible by 3? (a) 805 (b) 807 (c) 809 (d) 812
4. 4.Which of these numbers is divisible by 9? (a) 5698 (b) 5696 (c) 5697 (d) 5708
5. 5.What is the smallest number you must add to 114 to make it divisible by 8? (a) 5 (b) 7 (c) 6 (d) 2
6. 6.What is the remainder when 417 is divided by 10? (a) 7 (b) 6 (c) 3 (d) 8
7. 7.What is the smallest number you must add to 450 to make it divisible by 7? (a) 5 (b) 6 (c) 4 (d) 2
8. 8.What is the remainder when 398 is divided by 7? (a) 1 (b) 6 (c) 4 (d) 5
9. 9.What is the units digit of 8^49? (a) 2 (b) 6 (c) 8 (d) 4
10. 10.What is the units digit of 3^59? (a) 1 (b) 3 (c) 7 (d) 9
11. 11.What is the remainder when 167 is divided by 7? (a) 1 (b) 4 (c) 5 (d) 6
12. 12.What is the smallest number you must add to 552 to make it divisible by 7? (a) 3 (b) 1 (c) 2 (d) 6
13. 13.What is the remainder when 53 is divided by 4? (a) 2 (b) 3 (c) 0 (d) 1
14. 14.Which of these numbers is divisible by 6? (a) 4760 (b) 4768 (c) 4764 (d) 4775
15. 15.What is the units digit of 4^24? (a) 4 (b) 6 (c) 7 (d) 5
16. 16.What is the smallest number you must add to 345 to make it divisible by 9? (a) 7 (b) 6 (c) 5 (d) 3
17. 17.What is the smallest number you must add to 118 to make it divisible by 7? (a) 3 (b) 6 (c) 1 (d) 2
18. 18.What is the smallest number you must add to 381 to make it divisible by 7? (a) 5 (b) 6 (c) 3 (d) 4
19. 19.What is the remainder when 209 is divided by 6? (a) 3 (b) 4 (c) 5 (d) 1
20. 20.What is the units digit of 2^19? (a) 2 (b) 6 (c) 4 (d) 8
21. 21.What is the remainder when 410 is divided by 12? (a) 10 (b) 3 (c) 1 (d) 2
22. 22.What is the units digit of 2^60? (a) 2 (b) 8 (c) 6 (d) 4
23. 23.What is the remainder when 142 is divided by 3? (a) 2 (b) 0 (c) 1
24. 24.What is the smallest number you must add to 376 to make it divisible by 7? (a) 3 (b) 1 (c) 5 (d) 2
25. 25.What is the remainder when 154 is divided by 6? (a) 2 (b) 4 (c) 3 (d) 5
26. 26.Which of these numbers is divisible by 11? (a) 838 (b) 849 (c) 847 (d) 865
27. 27.What is the remainder when 442 is divided by 9? (a) 2 (b) 8 (c) 1 (d) 0
28. 28.What is the smallest number you must add to 533 to make it divisible by 9? (a) 2 (b) 6 (c) 7 (d) 8
29. 29.Which of these numbers is divisible by 3? (a) 1485 (b) 1484 (c) 1487 (d) 1489
30. 30.Which of these numbers is divisible by 4? (a) 3163 (b) 3153 (c) 3158 (d) 3156
31. 31.What is the units digit of 4^57? (a) 5 (b) 6 (c) 3 (d) 4
32. 32.Which of these numbers is divisible by 11? (a) 2349 (b) 2367 (c) 2354 (d) 2362
33. 33.What is the units digit of 7^25? (a) 3 (b) 1 (c) 9 (d) 7
34. 34.What is the units digit of 4^45? (a) 6 (b) 3 (c) 4 (d) 5
35. 35.What is the remainder when 385 is divided by 8? (a) 2 (b) 0 (c) 7 (d) 1
36. 36.What is the remainder when 233 is divided by 12? (a) 6 (b) 5 (c) 7 (d) 4
37. 37.Which of these numbers is divisible by 9? (a) 4788 (b) 4804 (c) 4789 (d) 4787
38. 38.What is the smallest number you must add to 68 to make it divisible by 3? (a) 2 (b) 1
39. 39.What is the smallest number you must add to 49 to make it divisible by 3? (a) 1 (b) 2
40. 40.What is the remainder when 70 is divided by 3? (a) 0 (b) 2 (c) 1

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