Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 408 · 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 3^46? (a) 3 (b) 7 (c) 1 (d) 9
2. 2.What is the smallest number you must add to 275 to make it divisible by 4? (a) 2 (b) 1 (c) 3
3. 3.What is the smallest number you must add to 182 to make it divisible by 4? (a) 1 (b) 3 (c) 2
4. 4.What is the remainder when 294 is divided by 12? (a) 5 (b) 8 (c) 7 (d) 6
5. 5.What is the remainder when 82 is divided by 8? (a) 6 (b) 2 (c) 1 (d) 3
6. 6.What is the smallest number you must add to 118 to make it divisible by 8? (a) 2 (b) 1 (c) 6 (d) 3
7. 7.What is the units digit of 8^45? (a) 8 (b) 6 (c) 4 (d) 2
8. 8.Which of these numbers is divisible by 11? (a) 4021 (b) 4026 (c) 4047 (d) 4031
9. 9.Which of these numbers is divisible by 3? (a) 2551 (b) 2545 (c) 2549 (d) 2547
10. 10.What is the units digit of 3^27? (a) 9 (b) 1 (c) 3 (d) 7
11. 11.What is the remainder when 314 is divided by 7? (a) 5 (b) 1 (c) 6 (d) 4
12. 12.What is the smallest number you must add to 179 to make it divisible by 6? (a) 3 (b) 1 (c) 5 (d) 2
13. 13.Which of these numbers is divisible by 6? (a) 1017 (b) 1014 (c) 1009 (d) 1021
14. 14.What is the units digit of 2^36? (a) 8 (b) 4 (c) 2 (d) 6
15. 15.What is the units digit of 3^14? (a) 7 (b) 9 (c) 3 (d) 1
16. 16.Which of these numbers is divisible by 8? (a) 321 (b) 320 (c) 316 (d) 332
17. 17.Which of these numbers is divisible by 9? (a) 5745 (b) 5726 (c) 5734 (d) 5733
18. 18.What is the units digit of 3^45? (a) 7 (b) 9 (c) 1 (d) 3
19. 19.What is the units digit of 9^46? (a) 9 (b) 2 (c) 1 (d) 0
20. 20.What is the smallest number you must add to 143 to make it divisible by 3? (a) 1 (b) 2
21. 21.What is the units digit of 4^24? (a) 6 (b) 4 (c) 7 (d) 5
22. 22.What is the smallest number you must add to 457 to make it divisible by 9? (a) 7 (b) 2 (c) 1 (d) 3
23. 23.Which of these numbers is divisible by 11? (a) 2413 (b) 2422 (c) 2434 (d) 2420
24. 24.What is the units digit of 9^14? (a) 9 (b) 0 (c) 1 (d) 2
25. 25.What is the units digit of 3^17? (a) 9 (b) 7 (c) 1 (d) 3
26. 26.What is the units digit of 4^37? (a) 6 (b) 3 (c) 4 (d) 5
27. 27.Which of these numbers is divisible by 4? (a) 1787 (b) 1778 (c) 1780 (d) 1781
28. 28.Which of these numbers is divisible by 6? (a) 2040 (b) 2037 (c) 2051 (d) 2042
29. 29.Which of these numbers is divisible by 4? (a) 730 (b) 725 (c) 728 (d) 735
30. 30.What is the units digit of 8^42? (a) 4 (b) 6 (c) 8 (d) 2
31. 31.What is the units digit of 3^35? (a) 7 (b) 3 (c) 9 (d) 1
32. 32.Which of these numbers is divisible by 6? (a) 2530 (b) 2523 (c) 2518 (d) 2520
33. 33.What is the remainder when 47 is divided by 3? (a) 2 (b) 1 (c) 0
34. 34.What is the remainder when 524 is divided by 7? (a) 4 (b) 6 (c) 5 (d) 1
35. 35.What is the units digit of 7^60? (a) 9 (b) 7 (c) 3 (d) 1
36. 36.What is the remainder when 245 is divided by 9? (a) 7 (b) 2 (c) 1 (d) 3
37. 37.What is the units digit of 8^36? (a) 2 (b) 8 (c) 4 (d) 6
38. 38.What is the remainder when 447 is divided by 8? (a) 1 (b) 6 (c) 5 (d) 7
39. 39.What is the remainder when 182 is divided by 3? (a) 1 (b) 2 (c) 0
40. 40.Which of these numbers is divisible by 3? (a) 1484 (b) 1487 (c) 1489 (d) 1485

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