Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 708 · 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.Which of these numbers is divisible by 6? (a) 1053 (b) 1059 (c) 1067 (d) 1056
2. 2.What is the remainder when 297 is divided by 5? (a) 3 (b) 1 (c) 2 (d) 4
3. 3.What is the smallest number you must add to 230 to make it divisible by 3? (a) 1 (b) 2
4. 4.What is the units digit of 3^36? (a) 7 (b) 9 (c) 3 (d) 1
5. 5.Which of these numbers is divisible by 3? (a) 2371 (b) 2368 (c) 2375 (d) 2370
6. 6.Which of these numbers is divisible by 11? (a) 2445 (b) 2431 (c) 2428 (d) 2437
7. 7.Which of these numbers is divisible by 6? (a) 964 (b) 954 (c) 955 (d) 953
8. 8.What is the units digit of 7^17? (a) 1 (b) 3 (c) 9 (d) 7
9. 9.What is the units digit of 2^27? (a) 8 (b) 2 (c) 4 (d) 6
10. 10.What is the smallest number you must add to 228 to make it divisible by 5? (a) 4 (b) 2 (c) 3 (d) 1
11. 11.What is the remainder when 190 is divided by 3? (a) 2 (b) 1 (c) 0
12. 12.What is the remainder when 350 is divided by 12? (a) 10 (b) 2 (c) 3 (d) 1
13. 13.Which of these numbers is divisible by 9? (a) 1456 (b) 1442 (c) 1440 (d) 1439
14. 14.What is the units digit of 4^33? (a) 4 (b) 5 (c) 3 (d) 6
15. 15.What is the smallest number you must add to 349 to make it divisible by 5? (a) 1 (b) 2 (c) 3 (d) 4
16. 16.What is the remainder when 111 is divided by 5? (a) 1 (b) 2 (c) 0 (d) 4
17. 17.What is the remainder when 439 is divided by 7? (a) 5 (b) 2 (c) 6 (d) 4
18. 18.Which of these numbers is divisible by 4? (a) 3282 (b) 3278 (c) 3280 (d) 3286
19. 19.What is the smallest number you must add to 412 to make it divisible by 11? (a) 8 (b) 7 (c) 6 (d) 5
20. 20.What is the smallest number you must add to 348 to make it divisible by 5? (a) 4 (b) 1 (c) 2 (d) 3
21. 21.What is the units digit of 3^38? (a) 3 (b) 7 (c) 1 (d) 9
22. 22.Which of these numbers is divisible by 11? (a) 5844 (b) 5830 (c) 5833 (d) 5824
23. 23.What is the units digit of 4^23? (a) 5 (b) 6 (c) 4 (d) 3
24. 24.Which of these numbers is divisible by 8? (a) 2698 (b) 2707 (c) 2696 (d) 2693
25. 25.What is the remainder when 206 is divided by 7? (a) 2 (b) 5 (c) 3 (d) 4
26. 26.Which of these numbers is divisible by 3? (a) 2561 (b) 2555 (c) 2556 (d) 2558
27. 27.Which of these numbers is divisible by 11? (a) 6141 (b) 6127 (c) 6132 (d) 6121
28. 28.Which of these numbers is divisible by 11? (a) 3648 (b) 3622 (c) 3635 (d) 3630
29. 29.What is the smallest number you must add to 389 to make it divisible by 9? (a) 6 (b) 8 (c) 7 (d) 2
30. 30.What is the units digit of 8^20? (a) 8 (b) 2 (c) 4 (d) 6
31. 31.Which of these numbers is divisible by 11? (a) 8380 (b) 8382 (c) 8394 (d) 8387
32. 32.Which of these numbers is divisible by 4? (a) 792 (b) 797 (c) 791 (d) 795
33. 33.What is the remainder when 747 is divided by 10? (a) 3 (b) 7 (c) 8 (d) 6
34. 34.Which of these numbers is divisible by 9? (a) 3649 (b) 3659 (c) 3666 (d) 3654
35. 35.What is the remainder when 62 is divided by 5? (a) 4 (b) 2 (c) 3 (d) 1
36. 36.Which of these numbers is divisible by 3? (a) 1141 (b) 1135 (c) 1138 (d) 1137
37. 37.Which of these numbers is divisible by 4? (a) 2819 (b) 2813 (c) 2816 (d) 2821
38. 38.What is the units digit of 2^60? (a) 8 (b) 4 (c) 2 (d) 6
39. 39.What is the smallest number you must add to 191 to make it divisible by 9? (a) 8 (b) 7 (c) 6 (d) 2
40. 40.What is the units digit of 4^13? (a) 3 (b) 5 (c) 4 (d) 6

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