Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 547 · 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 9^22? (a) 1 (b) 0 (c) 9 (d) 2
2. 2.What is the remainder when 229 is divided by 5? (a) 4 (b) 1 (c) 3 (d) 2
3. 3.Which of these numbers is divisible by 3? (a) 500 (b) 502 (c) 496 (d) 498
4. 4.What is the smallest number you must add to 355 to make it divisible by 6? (a) 4 (b) 3 (c) 1 (d) 5
5. 5.What is the units digit of 9^19? (a) 7 (b) 1 (c) 9 (d) 8
6. 6.What is the smallest number you must add to 259 to make it divisible by 5? (a) 2 (b) 4 (c) 3 (d) 1
7. 7.What is the units digit of 2^54? (a) 6 (b) 2 (c) 8 (d) 4
8. 8.What is the units digit of 9^32? (a) 9 (b) 2 (c) 1 (d) 0
9. 9.Which of these numbers is divisible by 9? (a) 2773 (b) 2755 (c) 2763 (d) 2768
10. 10.What is the remainder when 235 is divided by 10? (a) 7 (b) 5 (c) 4 (d) 6
11. 11.What is the smallest number you must add to 103 to make it divisible by 5? (a) 3 (b) 4 (c) 2 (d) 1
12. 12.What is the smallest number you must add to 608 to make it divisible by 11? (a) 3 (b) 7 (c) 8 (d) 9
13. 13.What is the remainder when 110 is divided by 3? (a) 0 (b) 1 (c) 2
14. 14.What is the smallest number you must add to 245 to make it divisible by 12? (a) 7 (b) 6 (c) 8 (d) 5
15. 15.Which of these numbers is divisible by 9? (a) 7170 (b) 7190 (c) 7175 (d) 7173
16. 16.Which of these numbers is divisible by 9? (a) 1184 (b) 1176 (c) 1168 (d) 1170
17. 17.What is the units digit of 7^18? (a) 9 (b) 1 (c) 3 (d) 7
18. 18.What is the smallest number you must add to 123 to make it divisible by 4? (a) 1 (b) 2 (c) 3
19. 19.What is the remainder when 175 is divided by 4? (a) 0 (b) 2 (c) 1 (d) 3
20. 20.What is the units digit of 7^38? (a) 1 (b) 9 (c) 3 (d) 7
21. 21.What is the smallest number you must add to 459 to make it divisible by 7? (a) 3 (b) 2 (c) 5 (d) 4
22. 22.What is the smallest number you must add to 438 to make it divisible by 7? (a) 3 (b) 5 (c) 4 (d) 2
23. 23.What is the remainder when 117 is divided by 5? (a) 2 (b) 4 (c) 1 (d) 3
24. 24.What is the remainder when 411 is divided by 8? (a) 2 (b) 3 (c) 4 (d) 5
25. 25.What is the smallest number you must add to 215 to make it divisible by 3? (a) 1 (b) 2
26. 26.What is the remainder when 122 is divided by 5? (a) 1 (b) 4 (c) 3 (d) 2
27. 27.Which of these numbers is divisible by 3? (a) 2183 (b) 2186 (c) 2184 (d) 2188
28. 28.What is the smallest number you must add to 713 to make it divisible by 12? (a) 7 (b) 5 (c) 6 (d) 8
29. 29.What is the units digit of 8^39? (a) 6 (b) 8 (c) 4 (d) 2
30. 30.What is the units digit of 7^39? (a) 9 (b) 3 (c) 7 (d) 1
31. 31.Which of these numbers is divisible by 6? (a) 3002 (b) 3009 (c) 3013 (d) 3006
32. 32.What is the units digit of 2^59? (a) 4 (b) 6 (c) 2 (d) 8
33. 33.What is the smallest number you must add to 224 to make it divisible by 11? (a) 8 (b) 7 (c) 6 (d) 4
34. 34.What is the remainder when 220 is divided by 6? (a) 3 (b) 4 (c) 2 (d) 5
35. 35.What is the remainder when 83 is divided by 3? (a) 0 (b) 1 (c) 2
36. 36.Which of these numbers is divisible by 4? (a) 473 (b) 481 (c) 478 (d) 476
37. 37.What is the remainder when 82 is divided by 8? (a) 1 (b) 6 (c) 2 (d) 3
38. 38.What is the smallest number you must add to 365 to make it divisible by 10? (a) 4 (b) 6 (c) 5 (d) 7
39. 39.Which of these numbers is divisible by 8? (a) 6313 (b) 6324 (c) 6329 (d) 6320
40. 40.Which of these numbers is divisible by 8? (a) 4719 (b) 4710 (c) 4712 (d) 4721

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