Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 602 · 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 147 is divided by 4? (a) 1 (b) 2 (c) 3 (d) 0
2. 2.What is the smallest number you must add to 338 to make it divisible by 10? (a) 1 (b) 3 (c) 2 (d) 8
3. 3.What is the smallest number you must add to 310 to make it divisible by 6? (a) 2 (b) 3 (c) 4 (d) 1
4. 4.What is the remainder when 806 is divided by 10? (a) 7 (b) 5 (c) 4 (d) 6
5. 5.Which of these numbers is divisible by 11? (a) 6457 (b) 6460 (c) 6471 (d) 6449
6. 6.Which of these numbers is divisible by 4? (a) 3147 (b) 3140 (c) 3137 (d) 3142
7. 7.What is the remainder when 241 is divided by 3? (a) 2 (b) 1 (c) 0
8. 8.What is the remainder when 235 is divided by 3? (a) 2 (b) 0 (c) 1
9. 9.What is the units digit of 8^11? (a) 6 (b) 4 (c) 8 (d) 2
10. 10.What is the smallest number you must add to 697 to make it divisible by 12? (a) 11 (b) 1 (c) 9 (d) 10
11. 11.Which of these numbers is divisible by 3? (a) 478 (b) 482 (c) 485 (d) 480
12. 12.Which of these numbers is divisible by 4? (a) 715 (b) 716 (c) 723 (d) 718
13. 13.What is the smallest number you must add to 963 to make it divisible by 12? (a) 8 (b) 10 (c) 3 (d) 9
14. 14.What is the units digit of 4^52? (a) 4 (b) 7 (c) 5 (d) 6
15. 15.What is the remainder when 645 is divided by 8? (a) 5 (b) 4 (c) 3 (d) 6
16. 16.Which of these numbers is divisible by 3? (a) 2018 (b) 2012 (c) 2013 (d) 2014
17. 17.What is the remainder when 548 is divided by 8? (a) 6 (b) 4 (c) 5 (d) 3
18. 18.What is the units digit of 2^54? (a) 8 (b) 2 (c) 6 (d) 4
19. 19.What is the units digit of 3^34? (a) 3 (b) 9 (c) 7 (d) 1
20. 20.What is the smallest number you must add to 194 to make it divisible by 11? (a) 5 (b) 3 (c) 4 (d) 7
21. 21.What is the remainder when 290 is divided by 6? (a) 4 (b) 2 (c) 1 (d) 3
22. 22.What is the units digit of 9^23? (a) 8 (b) 9 (c) 1 (d) 7
23. 23.What is the smallest number you must add to 250 to make it divisible by 8? (a) 2 (b) 7 (c) 5 (d) 6
24. 24.What is the smallest number you must add to 134 to make it divisible by 3? (a) 2 (b) 1
25. 25.What is the remainder when 386 is divided by 8? (a) 3 (b) 2 (c) 1 (d) 6
26. 26.Which of these numbers is divisible by 3? (a) 2221 (b) 2215 (c) 2219 (d) 2217
27. 27.What is the units digit of 3^31? (a) 3 (b) 1 (c) 7 (d) 9
28. 28.Which of these numbers is divisible by 11? (a) 8532 (b) 8545 (c) 8523 (d) 8525
29. 29.What is the remainder when 771 is divided by 12? (a) 2 (b) 3 (c) 9 (d) 4
30. 30.What is the remainder when 112 is divided by 5? (a) 3 (b) 1 (c) 4 (d) 2
31. 31.What is the units digit of 7^52? (a) 1 (b) 7 (c) 3 (d) 9
32. 32.Which of these numbers is divisible by 4? (a) 3425 (b) 3420 (c) 3421 (d) 3417
33. 33.What is the smallest number you must add to 54 to make it divisible by 4? (a) 2 (b) 3 (c) 1
34. 34.What is the smallest number you must add to 776 to make it divisible by 11? (a) 4 (b) 6 (c) 5 (d) 7
35. 35.Which of these numbers is divisible by 6? (a) 3361 (b) 3360 (c) 3359 (d) 3367
36. 36.What is the smallest number you must add to 235 to make it divisible by 3? (a) 1 (b) 2
37. 37.Which of these numbers is divisible by 9? (a) 1089 (b) 1088 (c) 1099 (d) 1094
38. 38.Which of these numbers is divisible by 11? (a) 620 (b) 627 (c) 635 (d) 640
39. 39.What is the units digit of 8^30? (a) 8 (b) 6 (c) 4 (d) 2
40. 40.Which of these numbers is divisible by 3? (a) 856 (b) 863 (c) 858 (d) 860

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