Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 535 · 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 8^33? (a) 2 (b) 4 (c) 6 (d) 8
2. 2.What is the smallest number you must add to 635 to make it divisible by 9? (a) 4 (b) 3 (c) 6 (d) 5
3. 3.Which of these numbers is divisible by 4? (a) 3542 (b) 3535 (c) 3536 (d) 3537
4. 4.What is the units digit of 8^39? (a) 8 (b) 4 (c) 2 (d) 6
5. 5.Which of these numbers is divisible by 6? (a) 3220 (b) 3213 (c) 3210 (d) 3209
6. 6.Which of these numbers is divisible by 8? (a) 1509 (b) 1514 (c) 1503 (d) 1504
7. 7.What is the smallest number you must add to 193 to make it divisible by 10? (a) 7 (b) 8 (c) 3 (d) 6
8. 8.What is the remainder when 405 is divided by 12? (a) 8 (b) 9 (c) 10 (d) 3
9. 9.Which of these numbers is divisible by 8? (a) 5300 (b) 5296 (c) 5306 (d) 5295
10. 10.What is the remainder when 212 is divided by 6? (a) 4 (b) 3 (c) 1 (d) 2
11. 11.What is the smallest number you must add to 202 to make it divisible by 3? (a) 1 (b) 2
12. 12.What is the units digit of 8^22? (a) 6 (b) 4 (c) 8 (d) 2
13. 13.What is the units digit of 8^60? (a) 4 (b) 6 (c) 8 (d) 2
14. 14.What is the smallest number you must add to 419 to make it divisible by 10? (a) 1 (b) 9 (c) 2 (d) 3
15. 15.Which of these numbers is divisible by 3? (a) 652 (b) 651 (c) 649 (d) 655
16. 16.What is the smallest number you must add to 870 to make it divisible by 12? (a) 6 (b) 5 (c) 7 (d) 8
17. 17.What is the smallest number you must add to 211 to make it divisible by 10? (a) 8 (b) 9 (c) 7 (d) 1
18. 18.Which of these numbers is divisible by 4? (a) 795 (b) 798 (c) 802 (d) 796
19. 19.What is the remainder when 250 is divided by 4? (a) 3 (b) 0 (c) 2 (d) 1
20. 20.What is the remainder when 157 is divided by 5? (a) 3 (b) 4 (c) 1 (d) 2
21. 21.What is the remainder when 499 is divided by 10? (a) 8 (b) 7 (c) 1 (d) 9
22. 22.What is the units digit of 9^40? (a) 2 (b) 9 (c) 0 (d) 1
23. 23.What is the smallest number you must add to 608 to make it divisible by 9? (a) 5 (b) 6 (c) 4 (d) 3
24. 24.What is the smallest number you must add to 76 to make it divisible by 3? (a) 1 (b) 2
25. 25.Which of these numbers is divisible by 9? (a) 1953 (b) 1958 (c) 1949 (d) 1970
26. 26.What is the remainder when 485 is divided by 6? (a) 4 (b) 1 (c) 3 (d) 5
27. 27.Which of these numbers is divisible by 9? (a) 5372 (b) 5362 (c) 5355 (d) 5347
28. 28.What is the remainder when 832 is divided by 12? (a) 8 (b) 4 (c) 3 (d) 5
29. 29.What is the smallest number you must add to 243 to make it divisible by 4? (a) 2 (b) 3 (c) 1
30. 30.Which of these numbers is divisible by 3? (a) 666 (b) 668 (c) 665 (d) 671
31. 31.What is the remainder when 836 is divided by 12? (a) 9 (b) 7 (c) 4 (d) 8
32. 32.What is the remainder when 449 is divided by 9? (a) 6 (b) 1 (c) 8 (d) 7
33. 33.What is the remainder when 274 is divided by 4? (a) 2 (b) 1 (c) 3 (d) 0
34. 34.What is the units digit of 9^23? (a) 7 (b) 9 (c) 8 (d) 1
35. 35.What is the remainder when 202 is divided by 3? (a) 0 (b) 2 (c) 1
36. 36.What is the units digit of 8^15? (a) 6 (b) 4 (c) 2 (d) 8
37. 37.Which of these numbers is divisible by 3? (a) 1534 (b) 1530 (c) 1532 (d) 1529
38. 38.What is the units digit of 3^45? (a) 9 (b) 7 (c) 1 (d) 3
39. 39.Which of these numbers is divisible by 11? (a) 1373 (b) 1393 (c) 1375 (d) 1382
40. 40.What is the units digit of 3^55? (a) 1 (b) 3 (c) 9 (d) 7

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