Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 363 · 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.Which of these numbers is divisible by 6? (a) 2736 (b) 2747 (c) 2734 (d) 2737
2. 2.What is the smallest number you must add to 343 to make it divisible by 10? (a) 7 (b) 6 (c) 3 (d) 8
3. 3.What is the remainder when 114 is divided by 5? (a) 1 (b) 2 (c) 4 (d) 3
4. 4.What is the smallest number you must add to 300 to make it divisible by 9? (a) 6 (b) 7 (c) 5 (d) 3
5. 5.Which of these numbers is divisible by 4? (a) 2982 (b) 2985 (c) 2980 (d) 2978
6. 6.What is the smallest number you must add to 334 to make it divisible by 9? (a) 6 (b) 7 (c) 1 (d) 8
7. 7.Which of these numbers is divisible by 9? (a) 6827 (b) 6811 (c) 6813 (d) 6818
8. 8.What is the smallest number you must add to 683 to make it divisible by 10? (a) 6 (b) 8 (c) 7 (d) 3
9. 9.Which of these numbers is divisible by 8? (a) 677 (b) 691 (c) 680 (d) 685
10. 10.Which of these numbers is divisible by 4? (a) 3027 (b) 3019 (c) 3020 (d) 3023
11. 11.Which of these numbers is divisible by 8? (a) 3276 (b) 3272 (c) 3284 (d) 3270
12. 12.What is the smallest number you must add to 203 to make it divisible by 3? (a) 1 (b) 2
13. 13.What is the smallest number you must add to 692 to make it divisible by 9? (a) 3 (b) 1 (c) 8 (d) 2
14. 14.What is the remainder when 192 is divided by 10? (a) 3 (b) 8 (c) 1 (d) 2
15. 15.What is the smallest number you must add to 106 to make it divisible by 4? (a) 2 (b) 1 (c) 3
16. 16.What is the units digit of 2^40? (a) 4 (b) 8 (c) 2 (d) 6
17. 17.What is the units digit of 9^19? (a) 8 (b) 9 (c) 1 (d) 7
18. 18.Which of these numbers is divisible by 11? (a) 9746 (b) 9767 (c) 9756 (d) 9745
19. 19.What is the units digit of 9^11? (a) 1 (b) 9 (c) 8 (d) 7
20. 20.Which of these numbers is divisible by 11? (a) 9417 (b) 9411 (c) 9416 (d) 9429
21. 21.What is the remainder when 340 is divided by 7? (a) 3 (b) 4 (c) 5 (d) 6
22. 22.What is the remainder when 379 is divided by 11? (a) 7 (b) 5 (c) 4 (d) 6
23. 23.Which of these numbers is divisible by 11? (a) 5464 (b) 5450 (c) 5477 (d) 5456
24. 24.Which of these numbers is divisible by 11? (a) 987 (b) 977 (c) 997 (d) 979
25. 25.What is the units digit of 8^41? (a) 8 (b) 2 (c) 6 (d) 4
26. 26.What is the remainder when 85 is divided by 3? (a) 1 (b) 2 (c) 0
27. 27.What is the remainder when 123 is divided by 4? (a) 1 (b) 3 (c) 0 (d) 2
28. 28.What is the units digit of 8^27? (a) 4 (b) 2 (c) 8 (d) 6
29. 29.Which of these numbers is divisible by 6? (a) 4345 (b) 4343 (c) 4354 (d) 4344
30. 30.What is the smallest number you must add to 525 to make it divisible by 8? (a) 4 (b) 2 (c) 3 (d) 5
31. 31.What is the smallest number you must add to 91 to make it divisible by 8? (a) 6 (b) 4 (c) 3 (d) 5
32. 32.What is the remainder when 346 is divided by 12? (a) 9 (b) 11 (c) 2 (d) 10
33. 33.What is the remainder when 221 is divided by 9? (a) 6 (b) 4 (c) 7 (d) 5
34. 34.What is the remainder when 149 is divided by 11? (a) 5 (b) 6 (c) 8 (d) 7
35. 35.What is the smallest number you must add to 127 to make it divisible by 7? (a) 6 (b) 1 (c) 4 (d) 5
36. 36.Which of these numbers is divisible by 11? (a) 5382 (b) 5409 (c) 5390 (d) 5394
37. 37.What is the units digit of 3^11? (a) 3 (b) 9 (c) 7 (d) 1
38. 38.What is the smallest number you must add to 117 to make it divisible by 8? (a) 2 (b) 4 (c) 3 (d) 5
39. 39.What is the units digit of 7^25? (a) 9 (b) 7 (c) 1 (d) 3
40. 40.What is the units digit of 8^16? (a) 6 (b) 2 (c) 8 (d) 4

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