Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 187 · 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 units digit of 7^36? (a) 1 (b) 9 (c) 3 (d) 7
2. 2.What is the remainder when 382 is divided by 5? (a) 1 (b) 2 (c) 4 (d) 3
3. 3.Which of these numbers is divisible by 3? (a) 2463 (b) 2462 (c) 2468 (d) 2464
4. 4.Which of these numbers is divisible by 11? (a) 3713 (b) 3695 (c) 3702 (d) 3696
5. 5.What is the smallest number you must add to 134 to make it divisible by 4? (a) 2 (b) 1 (c) 3
6. 6.What is the units digit of 2^24? (a) 4 (b) 2 (c) 6 (d) 8
7. 7.What is the units digit of 4^59? (a) 5 (b) 3 (c) 4 (d) 6
8. 8.What is the smallest number you must add to 158 to make it divisible by 12? (a) 9 (b) 2 (c) 10 (d) 11
9. 9.What is the smallest number you must add to 376 to make it divisible by 12? (a) 8 (b) 9 (c) 4 (d) 7
10. 10.What is the remainder when 247 is divided by 5? (a) 2 (b) 4 (c) 1 (d) 3
11. 11.What is the remainder when 246 is divided by 7? (a) 1 (b) 2 (c) 0 (d) 6
12. 12.What is the units digit of 4^13? (a) 3 (b) 4 (c) 6 (d) 5
13. 13.What is the remainder when 288 is divided by 7? (a) 0 (b) 6 (c) 1 (d) 2
14. 14.What is the remainder when 402 is divided by 8? (a) 6 (b) 3 (c) 1 (d) 2
15. 15.What is the remainder when 659 is divided by 10? (a) 9 (b) 1 (c) 8 (d) 7
16. 16.What is the units digit of 8^38? (a) 2 (b) 6 (c) 8 (d) 4
17. 17.What is the remainder when 249 is divided by 4? (a) 0 (b) 3 (c) 1 (d) 2
18. 18.What is the smallest number you must add to 74 to make it divisible by 3? (a) 1 (b) 2
19. 19.What is the remainder when 566 is divided by 12? (a) 1 (b) 3 (c) 2 (d) 10
20. 20.What is the smallest number you must add to 411 to make it divisible by 6? (a) 5 (b) 2 (c) 3 (d) 4
21. 21.Which of these numbers is divisible by 3? (a) 1956 (b) 1961 (c) 1954 (d) 1958
22. 22.What is the units digit of 4^41? (a) 4 (b) 3 (c) 5 (d) 6
23. 23.What is the units digit of 9^52? (a) 1 (b) 0 (c) 2 (d) 9
24. 24.What is the smallest number you must add to 348 to make it divisible by 5? (a) 2 (b) 1 (c) 3 (d) 4
25. 25.What is the remainder when 291 is divided by 4? (a) 0 (b) 2 (c) 3 (d) 1
26. 26.What is the units digit of 9^12? (a) 0 (b) 2 (c) 9 (d) 1
27. 27.What is the remainder when 610 is divided by 9? (a) 6 (b) 8 (c) 7 (d) 2
28. 28.What is the remainder when 159 is divided by 5? (a) 4 (b) 3 (c) 2 (d) 1
29. 29.What is the remainder when 454 is divided by 7? (a) 5 (b) 6 (c) 4 (d) 1
30. 30.What is the remainder when 473 is divided by 9? (a) 6 (b) 5 (c) 7 (d) 4
31. 31.What is the remainder when 651 is divided by 10? (a) 9 (b) 2 (c) 0 (d) 1
32. 32.What is the units digit of 3^47? (a) 9 (b) 7 (c) 3 (d) 1
33. 33.Which of these numbers is divisible by 11? (a) 5865 (b) 5874 (c) 5882 (d) 5887
34. 34.What is the units digit of 8^49? (a) 6 (b) 8 (c) 2 (d) 4
35. 35.Which of these numbers is divisible by 8? (a) 676 (b) 691 (c) 686 (d) 680
36. 36.What is the smallest number you must add to 290 to make it divisible by 12? (a) 11 (b) 9 (c) 2 (d) 10
37. 37.What is the smallest number you must add to 505 to make it divisible by 8? (a) 1 (b) 6 (c) 7 (d) 5
38. 38.Which of these numbers is divisible by 6? (a) 5116 (b) 5108 (c) 5106 (d) 5104
39. 39.Which of these numbers is divisible by 6? (a) 697 (b) 709 (c) 702 (d) 705
40. 40.What is the remainder when 171 is divided by 10? (a) 9 (b) 1 (c) 0 (d) 2

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