Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 980 · 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 9^36? (a) 0 (b) 2 (c) 1 (d) 9
2. 2.What is the units digit of 3^24? (a) 1 (b) 9 (c) 7 (d) 3
3. 3.Which of these numbers is divisible by 4? (a) 1468 (b) 1469 (c) 1473 (d) 1467
4. 4.What is the smallest number you must add to 157 to make it divisible by 4? (a) 3 (b) 2 (c) 1
5. 5.What is the smallest number you must add to 561 to make it divisible by 9? (a) 7 (b) 5 (c) 6 (d) 3
6. 6.What is the units digit of 2^54? (a) 4 (b) 6 (c) 8 (d) 2
7. 7.What is the remainder when 785 is divided by 12? (a) 7 (b) 4 (c) 5 (d) 6
8. 8.What is the units digit of 4^27? (a) 6 (b) 5 (c) 3 (d) 4
9. 9.Which of these numbers is divisible by 8? (a) 2094 (b) 2082 (c) 2098 (d) 2088
10. 10.What is the units digit of 8^33? (a) 6 (b) 4 (c) 8 (d) 2
11. 11.Which of these numbers is divisible by 6? (a) 338 (b) 330 (c) 327 (d) 333
12. 12.Which of these numbers is divisible by 9? (a) 1545 (b) 1538 (c) 1539 (d) 1551
13. 13.What is the smallest number you must add to 187 to make it divisible by 3? (a) 1 (b) 2
14. 14.What is the units digit of 9^40? (a) 9 (b) 2 (c) 1 (d) 0
15. 15.Which of these numbers is divisible by 6? (a) 3721 (b) 3718 (c) 3729 (d) 3720
16. 16.What is the units digit of 9^24? (a) 1 (b) 9 (c) 2 (d) 0
17. 17.What is the units digit of 8^21? (a) 6 (b) 2 (c) 4 (d) 8
18. 18.What is the remainder when 607 is divided by 9? (a) 4 (b) 3 (c) 5 (d) 6
19. 19.What is the units digit of 2^18? (a) 4 (b) 2 (c) 8 (d) 6
20. 20.What is the smallest number you must add to 462 to make it divisible by 12? (a) 5 (b) 7 (c) 6 (d) 8
21. 21.What is the remainder when 272 is divided by 6? (a) 1 (b) 3 (c) 2 (d) 4
22. 22.What is the remainder when 455 is divided by 10? (a) 7 (b) 4 (c) 6 (d) 5
23. 23.What is the smallest number you must add to 211 to make it divisible by 4? (a) 1 (b) 3 (c) 2
24. 24.What is the units digit of 2^52? (a) 2 (b) 6 (c) 4 (d) 8
25. 25.Which of these numbers is divisible by 11? (a) 9739 (b) 9764 (c) 9754 (d) 9746
26. 26.Which of these numbers is divisible by 11? (a) 7073 (b) 7069 (c) 7079 (d) 7090
27. 27.What is the smallest number you must add to 203 to make it divisible by 3? (a) 2 (b) 1
28. 28.What is the smallest number you must add to 58 to make it divisible by 5? (a) 4 (b) 2 (c) 1 (d) 3
29. 29.What is the units digit of 9^42? (a) 0 (b) 1 (c) 9 (d) 2
30. 30.What is the smallest number you must add to 266 to make it divisible by 6? (a) 4 (b) 5 (c) 2 (d) 3
31. 31.Which of these numbers is divisible by 8? (a) 4962 (b) 4960 (c) 4971 (d) 4957
32. 32.What is the smallest number you must add to 300 to make it divisible by 7? (a) 6 (b) 1 (c) 3 (d) 2
33. 33.What is the units digit of 7^33? (a) 9 (b) 7 (c) 1 (d) 3
34. 34.Which of these numbers is divisible by 8? (a) 2185 (b) 2178 (c) 2184 (d) 2197
35. 35.What is the smallest number you must add to 109 to make it divisible by 8? (a) 4 (b) 5 (c) 2 (d) 3
36. 36.Which of these numbers is divisible by 3? (a) 426 (b) 431 (c) 424 (d) 427
37. 37.Which of these numbers is divisible by 3? (a) 2411 (b) 2416 (c) 2414 (d) 2412
38. 38.What is the units digit of 2^43? (a) 4 (b) 2 (c) 6 (d) 8
39. 39.What is the units digit of 7^51? (a) 7 (b) 1 (c) 9 (d) 3
40. 40.What is the remainder when 732 is divided by 10? (a) 8 (b) 1 (c) 3 (d) 2

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