Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 889 · 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 9? (a) 1380 (b) 1389 (c) 1386 (d) 1396
2. 2.What is the units digit of 7^47? (a) 9 (b) 3 (c) 7 (d) 1
3. 3.What is the remainder when 275 is divided by 4? (a) 0 (b) 1 (c) 3 (d) 2
4. 4.What is the smallest number you must add to 688 to make it divisible by 10? (a) 8 (b) 3 (c) 1 (d) 2
5. 5.What is the units digit of 4^39? (a) 4 (b) 5 (c) 3 (d) 6
6. 6.What is the smallest number you must add to 418 to make it divisible by 9? (a) 7 (b) 4 (c) 5 (d) 6
7. 7.What is the remainder when 109 is divided by 8? (a) 4 (b) 3 (c) 5 (d) 6
8. 8.What is the units digit of 7^56? (a) 9 (b) 1 (c) 7 (d) 3
9. 9.What is the units digit of 9^47? (a) 9 (b) 1 (c) 8 (d) 7
10. 10.What is the units digit of 7^45? (a) 3 (b) 7 (c) 1 (d) 9
11. 11.Which of these numbers is divisible by 8? (a) 6939 (b) 6948 (c) 6933 (d) 6936
12. 12.What is the smallest number you must add to 650 to make it divisible by 9? (a) 8 (b) 6 (c) 2 (d) 7
13. 13.What is the remainder when 233 is divided by 3? (a) 2 (b) 0 (c) 1
14. 14.What is the smallest number you must add to 334 to make it divisible by 7? (a) 5 (b) 3 (c) 2 (d) 1
15. 15.What is the smallest number you must add to 556 to make it divisible by 7? (a) 6 (b) 4 (c) 3 (d) 5
16. 16.What is the smallest number you must add to 178 to make it divisible by 3? (a) 1 (b) 2
17. 17.What is the units digit of 2^15? (a) 8 (b) 2 (c) 4 (d) 6
18. 18.Which of these numbers is divisible by 6? (a) 4846 (b) 4842 (c) 4841 (d) 4853
19. 19.What is the remainder when 498 is divided by 11? (a) 3 (b) 4 (c) 2 (d) 8
20. 20.Which of these numbers is divisible by 6? (a) 4255 (b) 4265 (c) 4249 (d) 4254
21. 21.What is the smallest number you must add to 89 to make it divisible by 3? (a) 2 (b) 1
22. 22.What is the remainder when 162 is divided by 7? (a) 6 (b) 1 (c) 2 (d) 0
23. 23.What is the smallest number you must add to 637 to make it divisible by 9? (a) 1 (b) 7 (c) 2 (d) 3
24. 24.What is the remainder when 362 is divided by 7? (a) 6 (b) 4 (c) 2 (d) 5
25. 25.What is the smallest number you must add to 335 to make it divisible by 7? (a) 1 (b) 3 (c) 2 (d) 6
26. 26.Which of these numbers is divisible by 8? (a) 4559 (b) 4550 (c) 4561 (d) 4552
27. 27.What is the units digit of 9^11? (a) 9 (b) 7 (c) 1 (d) 8
28. 28.What is the smallest number you must add to 281 to make it divisible by 5? (a) 1 (b) 3 (c) 2 (d) 4
29. 29.Which of these numbers is divisible by 9? (a) 627 (b) 638 (c) 618 (d) 621
30. 30.What is the smallest number you must add to 107 to make it divisible by 6? (a) 2 (b) 3 (c) 5 (d) 1
31. 31.Which of these numbers is divisible by 4? (a) 508 (b) 507 (c) 515 (d) 510
32. 32.What is the units digit of 7^60? (a) 7 (b) 9 (c) 3 (d) 1
33. 33.What is the units digit of 9^14? (a) 1 (b) 9 (c) 2 (d) 0
34. 34.What is the units digit of 9^16? (a) 0 (b) 2 (c) 1 (d) 9
35. 35.What is the units digit of 4^33? (a) 4 (b) 6 (c) 3 (d) 5
36. 36.What is the smallest number you must add to 156 to make it divisible by 7? (a) 4 (b) 5 (c) 2 (d) 6
37. 37.What is the remainder when 94 is divided by 4? (a) 0 (b) 2 (c) 1 (d) 3
38. 38.Which of these numbers is divisible by 9? (a) 6593 (b) 6597 (c) 6607 (d) 6605
39. 39.Which of these numbers is divisible by 4? (a) 1362 (b) 1355 (c) 1356 (d) 1359
40. 40.What is the units digit of 4^24? (a) 6 (b) 5 (c) 7 (d) 4

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