Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 467 · 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 remainder when 450 is divided by 11? (a) 1 (b) 8 (c) 9 (d) 10
2. 2.Which of these numbers is divisible by 4? (a) 638 (b) 633 (c) 641 (d) 636
3. 3.What is the smallest number you must add to 469 to make it divisible by 6? (a) 4 (b) 3 (c) 5 (d) 1
4. 4.Which of these numbers is divisible by 9? (a) 2679 (b) 2673 (c) 2668 (d) 2683
5. 5.What is the smallest number you must add to 285 to make it divisible by 10? (a) 5 (b) 6 (c) 4 (d) 7
6. 6.What is the smallest number you must add to 72 to make it divisible by 5? (a) 1 (b) 4 (c) 3 (d) 2
7. 7.Which of these numbers is divisible by 3? (a) 2373 (b) 2371 (c) 2377 (d) 2374
8. 8.What is the remainder when 446 is divided by 10? (a) 5 (b) 6 (c) 4 (d) 7
9. 9.What is the units digit of 7^53? (a) 7 (b) 3 (c) 1 (d) 9
10. 10.Which of these numbers is divisible by 11? (a) 1418 (b) 1398 (c) 1397 (d) 1392
11. 11.What is the remainder when 107 is divided by 3? (a) 1 (b) 2 (c) 0
12. 12.Which of these numbers is divisible by 4? (a) 2797 (b) 2795 (c) 2789 (d) 2792
13. 13.Which of these numbers is divisible by 9? (a) 4238 (b) 4246 (c) 4252 (d) 4239
14. 14.What is the units digit of 3^47? (a) 9 (b) 1 (c) 3 (d) 7
15. 15.What is the remainder when 294 is divided by 4? (a) 1 (b) 0 (c) 3 (d) 2
16. 16.Which of these numbers is divisible by 9? (a) 6060 (b) 6069 (c) 6055 (d) 6057
17. 17.What is the remainder when 239 is divided by 8? (a) 7 (b) 1 (c) 5 (d) 6
18. 18.What is the remainder when 615 is divided by 8? (a) 6 (b) 1 (c) 5 (d) 7
19. 19.Which of these numbers is divisible by 3? (a) 2413 (b) 2417 (c) 2412 (d) 2411
20. 20.Which of these numbers is divisible by 9? (a) 1322 (b) 1330 (c) 1307 (d) 1314
21. 21.What is the smallest number you must add to 468 to make it divisible by 10? (a) 1 (b) 8 (c) 2 (d) 3
22. 22.What is the units digit of 8^43? (a) 8 (b) 4 (c) 6 (d) 2
23. 23.What is the remainder when 194 is divided by 3? (a) 0 (b) 2 (c) 1
24. 24.Which of these numbers is divisible by 9? (a) 1727 (b) 1710 (c) 1708 (d) 1714
25. 25.What is the smallest number you must add to 87 to make it divisible by 8? (a) 3 (b) 2 (c) 1 (d) 7
26. 26.What is the remainder when 103 is divided by 5? (a) 4 (b) 3 (c) 1 (d) 2
27. 27.Which of these numbers is divisible by 3? (a) 750 (b) 754 (c) 751 (d) 748
28. 28.What is the remainder when 277 is divided by 8? (a) 6 (b) 4 (c) 3 (d) 5
29. 29.Which of these numbers is divisible by 11? (a) 6660 (b) 6668 (c) 6655 (d) 6647
30. 30.What is the smallest number you must add to 747 to make it divisible by 11? (a) 1 (b) 2 (c) 3 (d) 10
31. 31.What is the smallest number you must add to 447 to make it divisible by 11? (a) 3 (b) 7 (c) 4 (d) 5
32. 32.What is the units digit of 7^41? (a) 7 (b) 9 (c) 1 (d) 3
33. 33.What is the units digit of 7^59? (a) 1 (b) 3 (c) 9 (d) 7
34. 34.What is the units digit of 8^26? (a) 4 (b) 2 (c) 8 (d) 6
35. 35.What is the smallest number you must add to 807 to make it divisible by 10? (a) 4 (b) 2 (c) 7 (d) 3
36. 36.Which of these numbers is divisible by 6? (a) 478 (b) 472 (c) 481 (d) 474
37. 37.Which of these numbers is divisible by 6? (a) 842 (b) 834 (c) 838 (d) 832
38. 38.Which of these numbers is divisible by 3? (a) 2017 (b) 2015 (c) 2020 (d) 2016
39. 39.What is the units digit of 8^58? (a) 8 (b) 2 (c) 4 (d) 6
40. 40.What is the smallest number you must add to 323 to make it divisible by 4? (a) 1 (b) 3 (c) 2

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