Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 125 · 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 209 is divided by 7? (a) 1 (b) 5 (c) 6 (d) 4
2. 2.What is the smallest number you must add to 215 to make it divisible by 8? (a) 1 (b) 3 (c) 7 (d) 2
3. 3.What is the remainder when 225 is divided by 4? (a) 2 (b) 0 (c) 3 (d) 1
4. 4.Which of these numbers is divisible by 4? (a) 316 (b) 317 (c) 313 (d) 323
5. 5.What is the units digit of 7^30? (a) 9 (b) 1 (c) 7 (d) 3
6. 6.What is the remainder when 69 is divided by 5? (a) 2 (b) 4 (c) 3 (d) 1
7. 7.What is the remainder when 157 is divided by 5? (a) 4 (b) 2 (c) 3 (d) 1
8. 8.Which of these numbers is divisible by 3? (a) 2562 (b) 2567 (c) 2560 (d) 2563
9. 9.What is the remainder when 312 is divided by 7? (a) 5 (b) 3 (c) 6 (d) 4
10. 10.Which of these numbers is divisible by 3? (a) 1849 (b) 1852 (c) 1856 (d) 1851
11. 11.Which of these numbers is divisible by 11? (a) 5379 (b) 5395 (c) 5385 (d) 5369
12. 12.What is the smallest number you must add to 152 to make it divisible by 12? (a) 3 (b) 8 (c) 4 (d) 5
13. 13.What is the units digit of 2^14? (a) 6 (b) 8 (c) 2 (d) 4
14. 14.What is the remainder when 212 is divided by 5? (a) 1 (b) 4 (c) 2 (d) 3
15. 15.Which of these numbers is divisible by 3? (a) 1979 (b) 1981 (c) 1985 (d) 1980
16. 16.What is the remainder when 541 is divided by 8? (a) 4 (b) 6 (c) 5 (d) 3
17. 17.What is the units digit of 2^39? (a) 8 (b) 6 (c) 4 (d) 2
18. 18.What is the smallest number you must add to 207 to make it divisible by 11? (a) 1 (b) 9 (c) 2 (d) 3
19. 19.What is the units digit of 7^18? (a) 1 (b) 3 (c) 9 (d) 7
20. 20.What is the smallest number you must add to 194 to make it divisible by 4? (a) 2 (b) 3 (c) 1
21. 21.Which of these numbers is divisible by 4? (a) 3371 (b) 3373 (c) 3379 (d) 3372
22. 22.What is the units digit of 7^19? (a) 7 (b) 3 (c) 1 (d) 9
23. 23.What is the remainder when 174 is divided by 12? (a) 6 (b) 7 (c) 5 (d) 8
24. 24.Which of these numbers is divisible by 6? (a) 560 (b) 553 (c) 568 (d) 558
25. 25.What is the remainder when 677 is divided by 11? (a) 5 (b) 6 (c) 7 (d) 8
26. 26.What is the smallest number you must add to 236 to make it divisible by 10? (a) 3 (b) 6 (c) 5 (d) 4
27. 27.What is the remainder when 664 is divided by 10? (a) 4 (b) 6 (c) 5 (d) 3
28. 28.What is the remainder when 194 is divided by 6? (a) 1 (b) 4 (c) 2 (d) 3
29. 29.Which of these numbers is divisible by 3? (a) 1151 (b) 1154 (c) 1157 (d) 1152
30. 30.What is the remainder when 839 is divided by 11? (a) 8 (b) 4 (c) 2 (d) 3
31. 31.Which of these numbers is divisible by 11? (a) 8672 (b) 8651 (c) 8657 (d) 8662
32. 32.What is the smallest number you must add to 367 to make it divisible by 9? (a) 3 (b) 7 (c) 2 (d) 1
33. 33.What is the units digit of 8^27? (a) 8 (b) 4 (c) 6 (d) 2
34. 34.What is the remainder when 463 is divided by 6? (a) 5 (b) 0 (c) 2 (d) 1
35. 35.What is the smallest number you must add to 431 to make it divisible by 10? (a) 9 (b) 8 (c) 1 (d) 7
36. 36.What is the remainder when 52 is divided by 3? (a) 1 (b) 0 (c) 2
37. 37.Which of these numbers is divisible by 8? (a) 4613 (b) 4616 (c) 4618 (d) 4626
38. 38.What is the remainder when 503 is divided by 9? (a) 8 (b) 1 (c) 6 (d) 7
39. 39.Which of these numbers is divisible by 6? (a) 2053 (b) 2060 (c) 2052 (d) 2050
40. 40.What is the smallest number you must add to 468 to make it divisible by 10? (a) 3 (b) 8 (c) 2 (d) 1

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