Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 902 · 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 7^29? (a) 7 (b) 3 (c) 1 (d) 9
2. 2.What is the smallest number you must add to 367 to make it divisible by 8? (a) 1 (b) 7 (c) 3 (d) 2
3. 3.Which of these numbers is divisible by 6? (a) 1682 (b) 1694 (c) 1687 (d) 1686
4. 4.What is the units digit of 4^18? (a) 6 (b) 4 (c) 7 (d) 5
5. 5.What is the remainder when 215 is divided by 11? (a) 7 (b) 6 (c) 5 (d) 8
6. 6.What is the smallest number you must add to 214 to make it divisible by 4? (a) 2 (b) 1 (c) 3
7. 7.Which of these numbers is divisible by 3? (a) 2071 (b) 2065 (c) 2068 (d) 2067
8. 8.What is the remainder when 133 is divided by 4? (a) 3 (b) 1 (c) 2 (d) 0
9. 9.Which of these numbers is divisible by 11? (a) 2881 (b) 2860 (c) 2869 (d) 2850
10. 10.Which of these numbers is divisible by 4? (a) 402 (b) 400 (c) 405 (d) 398
11. 11.What is the remainder when 96 is divided by 7? (a) 2 (b) 5 (c) 6 (d) 4
12. 12.What is the remainder when 332 is divided by 11? (a) 1 (b) 3 (c) 9 (d) 2
13. 13.What is the remainder when 203 is divided by 3? (a) 0 (b) 2 (c) 1
14. 14.What is the smallest number you must add to 266 to make it divisible by 11? (a) 10 (b) 8 (c) 2 (d) 9
15. 15.What is the smallest number you must add to 68 to make it divisible by 5? (a) 4 (b) 2 (c) 3 (d) 1
16. 16.What is the units digit of 8^55? (a) 4 (b) 8 (c) 2 (d) 6
17. 17.Which of these numbers is divisible by 4? (a) 538 (b) 542 (c) 536 (d) 533
18. 18.What is the units digit of 4^10? (a) 7 (b) 4 (c) 6 (d) 5
19. 19.What is the remainder when 509 is divided by 9? (a) 7 (b) 5 (c) 4 (d) 6
20. 20.What is the units digit of 3^57? (a) 1 (b) 7 (c) 3 (d) 9
21. 21.Which of these numbers is divisible by 4? (a) 846 (b) 841 (c) 844 (d) 850
22. 22.Which of these numbers is divisible by 8? (a) 6763 (b) 6782 (c) 6775 (d) 6768
23. 23.What is the remainder when 524 is divided by 11? (a) 8 (b) 6 (c) 4 (d) 7
24. 24.What is the smallest number you must add to 251 to make it divisible by 6? (a) 1 (b) 2 (c) 5 (d) 3
25. 25.Which of these numbers is divisible by 4? (a) 3471 (b) 3465 (c) 3464 (d) 3461
26. 26.What is the units digit of 7^12? (a) 1 (b) 3 (c) 9 (d) 7
27. 27.Which of these numbers is divisible by 6? (a) 598 (b) 611 (c) 603 (d) 600
28. 28.Which of these numbers is divisible by 11? (a) 6828 (b) 6840 (c) 6844 (d) 6831
29. 29.Which of these numbers is divisible by 4? (a) 2879 (b) 2870 (c) 2872 (d) 2873
30. 30.Which of these numbers is divisible by 8? (a) 2102 (b) 2109 (c) 2094 (d) 2096
31. 31.What is the smallest number you must add to 202 to make it divisible by 10? (a) 2 (b) 9 (c) 7 (d) 8
32. 32.What is the remainder when 746 is divided by 10? (a) 7 (b) 6 (c) 5 (d) 4
33. 33.What is the smallest number you must add to 301 to make it divisible by 4? (a) 1 (b) 2 (c) 3
34. 34.What is the units digit of 2^18? (a) 6 (b) 4 (c) 8 (d) 2
35. 35.Which of these numbers is divisible by 4? (a) 1162 (b) 1165 (c) 1169 (d) 1164
36. 36.Which of these numbers is divisible by 3? (a) 2448 (b) 2447 (c) 2449 (d) 2452
37. 37.What is the units digit of 9^24? (a) 2 (b) 9 (c) 0 (d) 1
38. 38.What is the remainder when 495 is divided by 12? (a) 2 (b) 4 (c) 3 (d) 9
39. 39.What is the remainder when 185 is divided by 12? (a) 5 (b) 4 (c) 7 (d) 6
40. 40.What is the remainder when 396 is divided by 10? (a) 4 (b) 7 (c) 5 (d) 6

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