Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 579 · 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 smallest number you must add to 394 to make it divisible by 9? (a) 3 (b) 2 (c) 7 (d) 1
2. 2.Which of these numbers is divisible by 3? (a) 1508 (b) 1510 (c) 1514 (d) 1509
3. 3.What is the remainder when 201 is divided by 10? (a) 2 (b) 1 (c) 9 (d) 0
4. 4.What is the smallest number you must add to 159 to make it divisible by 11? (a) 8 (b) 6 (c) 5 (d) 7
5. 5.What is the smallest number you must add to 338 to make it divisible by 11? (a) 8 (b) 4 (c) 2 (d) 3
6. 6.What is the units digit of 3^19? (a) 3 (b) 7 (c) 9 (d) 1
7. 7.Which of these numbers is divisible by 6? (a) 3920 (b) 3910 (c) 3912 (d) 3914
8. 8.Which of these numbers is divisible by 11? (a) 3641 (b) 3656 (c) 3648 (d) 3632
9. 9.What is the smallest number you must add to 578 to make it divisible by 9? (a) 8 (b) 6 (c) 2 (d) 7
10. 10.Which of these numbers is divisible by 9? (a) 6231 (b) 6227 (c) 6244 (d) 6228
11. 11.Which of these numbers is divisible by 8? (a) 4304 (b) 4299 (c) 4305 (d) 4319
12. 12.What is the remainder when 229 is divided by 10? (a) 8 (b) 7 (c) 1 (d) 9
13. 13.What is the smallest number you must add to 861 to make it divisible by 11? (a) 3 (b) 9 (c) 7 (d) 8
14. 14.What is the smallest number you must add to 206 to make it divisible by 5? (a) 4 (b) 3 (c) 2 (d) 1
15. 15.What is the remainder when 373 is divided by 8? (a) 3 (b) 5 (c) 4 (d) 6
16. 16.What is the remainder when 85 is divided by 3? (a) 1 (b) 2 (c) 0
17. 17.Which of these numbers is divisible by 3? (a) 805 (b) 808 (c) 811 (d) 807
18. 18.Which of these numbers is divisible by 4? (a) 851 (b) 847 (c) 854 (d) 848
19. 19.What is the units digit of 4^37? (a) 6 (b) 4 (c) 5 (d) 3
20. 20.What is the remainder when 389 is divided by 6? (a) 5 (b) 3 (c) 1 (d) 4
21. 21.What is the units digit of 3^40? (a) 1 (b) 3 (c) 7 (d) 9
22. 22.What is the units digit of 4^20? (a) 5 (b) 6 (c) 4 (d) 7
23. 23.What is the smallest number you must add to 101 to make it divisible by 5? (a) 2 (b) 1 (c) 4 (d) 3
24. 24.What is the remainder when 429 is divided by 7? (a) 2 (b) 1 (c) 3 (d) 5
25. 25.Which of these numbers is divisible by 9? (a) 5612 (b) 5606 (c) 5607 (d) 5618
26. 26.Which of these numbers is divisible by 6? (a) 491 (b) 494 (c) 486 (d) 485
27. 27.Which of these numbers is divisible by 8? (a) 2953 (b) 2969 (c) 2960 (d) 2966
28. 28.Which of these numbers is divisible by 11? (a) 3260 (b) 3267 (c) 3285 (d) 3270
29. 29.Which of these numbers is divisible by 11? (a) 2452 (b) 2441 (c) 2428 (d) 2431
30. 30.What is the remainder when 679 is divided by 9? (a) 4 (b) 5 (c) 6 (d) 3
31. 31.What is the units digit of 7^43? (a) 1 (b) 3 (c) 7 (d) 9
32. 32.What is the units digit of 8^56? (a) 8 (b) 4 (c) 6 (d) 2
33. 33.What is the smallest number you must add to 38 to make it divisible by 3? (a) 2 (b) 1
34. 34.What is the units digit of 9^37? (a) 7 (b) 9 (c) 1 (d) 8
35. 35.What is the smallest number you must add to 171 to make it divisible by 5? (a) 4 (b) 2 (c) 1 (d) 3
36. 36.What is the units digit of 9^26? (a) 9 (b) 0 (c) 1 (d) 2
37. 37.What is the remainder when 514 is divided by 9? (a) 1 (b) 2 (c) 8 (d) 0
38. 38.Which of these numbers is divisible by 3? (a) 604 (b) 602 (c) 600 (d) 599
39. 39.Which of these numbers is divisible by 9? (a) 6722 (b) 6735 (c) 6730 (d) 6723
40. 40.Which of these numbers is divisible by 11? (a) 1166 (b) 1172 (c) 1184 (d) 1158

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