Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 165 · 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 224 is divided by 3? (a) 0 (b) 1 (c) 2
2. 2.What is the units digit of 3^14? (a) 9 (b) 7 (c) 1 (d) 3
3. 3.What is the units digit of 9^52? (a) 9 (b) 0 (c) 2 (d) 1
4. 4.What is the smallest number you must add to 521 to make it divisible by 7? (a) 6 (b) 4 (c) 3 (d) 5
5. 5.What is the units digit of 3^34? (a) 9 (b) 3 (c) 1 (d) 7
6. 6.What is the smallest number you must add to 237 to make it divisible by 4? (a) 1 (b) 2 (c) 3
7. 7.What is the units digit of 2^20? (a) 4 (b) 8 (c) 2 (d) 6
8. 8.Which of these numbers is divisible by 4? (a) 376 (b) 375 (c) 381 (d) 377
9. 9.What is the remainder when 169 is divided by 3? (a) 1 (b) 0 (c) 2
10. 10.Which of these numbers is divisible by 9? (a) 3936 (b) 3957 (c) 3942 (d) 3949
11. 11.What is the smallest number you must add to 196 to make it divisible by 12? (a) 7 (b) 4 (c) 9 (d) 8
12. 12.What is the smallest number you must add to 333 to make it divisible by 12? (a) 9 (b) 3 (c) 2 (d) 4
13. 13.What is the units digit of 3^55? (a) 9 (b) 7 (c) 1 (d) 3
14. 14.What is the smallest number you must add to 310 to make it divisible by 4? (a) 1 (b) 3 (c) 2
15. 15.What is the smallest number you must add to 659 to make it divisible by 11? (a) 3 (b) 1 (c) 2 (d) 10
16. 16.What is the smallest number you must add to 47 to make it divisible by 4? (a) 1 (b) 3 (c) 2
17. 17.What is the units digit of 7^20? (a) 9 (b) 1 (c) 7 (d) 3
18. 18.Which of these numbers is divisible by 9? (a) 2744 (b) 2745 (c) 2761 (d) 2746
19. 19.What is the units digit of 3^18? (a) 3 (b) 7 (c) 1 (d) 9
20. 20.Which of these numbers is divisible by 6? (a) 2230 (b) 2233 (c) 2222 (d) 2226
21. 21.What is the smallest number you must add to 201 to make it divisible by 4? (a) 1 (b) 2 (c) 3
22. 22.What is the remainder when 77 is divided by 6? (a) 1 (b) 4 (c) 3 (d) 5
23. 23.Which of these numbers is divisible by 3? (a) 466 (b) 460 (c) 462 (d) 463
24. 24.What is the units digit of 7^19? (a) 3 (b) 9 (c) 1 (d) 7
25. 25.What is the smallest number you must add to 241 to make it divisible by 9? (a) 1 (b) 3 (c) 7 (d) 2
26. 26.Which of these numbers is divisible by 3? (a) 1630 (b) 1625 (c) 1626 (d) 1627
27. 27.What is the units digit of 8^12? (a) 8 (b) 2 (c) 6 (d) 4
28. 28.What is the units digit of 3^16? (a) 9 (b) 3 (c) 7 (d) 1
29. 29.What is the units digit of 7^18? (a) 7 (b) 1 (c) 3 (d) 9
30. 30.What is the smallest number you must add to 150 to make it divisible by 4? (a) 2 (b) 3 (c) 1
31. 31.What is the units digit of 8^11? (a) 6 (b) 2 (c) 8 (d) 4
32. 32.What is the remainder when 235 is divided by 6? (a) 1 (b) 0 (c) 2 (d) 5
33. 33.What is the units digit of 7^52? (a) 9 (b) 3 (c) 7 (d) 1
34. 34.Which of these numbers is divisible by 4? (a) 3417 (b) 3415 (c) 3416 (d) 3421
35. 35.What is the smallest number you must add to 495 to make it divisible by 7? (a) 5 (b) 1 (c) 2 (d) 3
36. 36.What is the units digit of 3^52? (a) 7 (b) 3 (c) 1 (d) 9
37. 37.What is the units digit of 2^55? (a) 8 (b) 2 (c) 4 (d) 6
38. 38.What is the units digit of 3^17? (a) 9 (b) 1 (c) 3 (d) 7
39. 39.What is the remainder when 139 is divided by 3? (a) 0 (b) 1 (c) 2
40. 40.What is the smallest number you must add to 614 to make it divisible by 9? (a) 7 (b) 6 (c) 8 (d) 2

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