Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 500 · 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 4^41? (a) 6 (b) 4 (c) 3 (d) 5
2. 2.Which of these numbers is divisible by 4? (a) 3353 (b) 3356 (c) 3357 (d) 3361
3. 3.What is the units digit of 3^28? (a) 7 (b) 3 (c) 9 (d) 1
4. 4.Which of these numbers is divisible by 3? (a) 387 (b) 389 (c) 386 (d) 392
5. 5.What is the remainder when 251 is divided by 10? (a) 0 (b) 1 (c) 9 (d) 2
6. 6.What is the remainder when 328 is divided by 11? (a) 8 (b) 2 (c) 10 (d) 9
7. 7.What is the smallest number you must add to 89 to make it divisible by 3? (a) 2 (b) 1
8. 8.What is the smallest number you must add to 812 to make it divisible by 11? (a) 9 (b) 1 (c) 2 (d) 3
9. 9.What is the smallest number you must add to 760 to make it divisible by 11? (a) 8 (b) 9 (c) 1 (d) 10
10. 10.What is the smallest number you must add to 442 to make it divisible by 8? (a) 2 (b) 5 (c) 6 (d) 7
11. 11.What is the remainder when 129 is divided by 11? (a) 7 (b) 9 (c) 8 (d) 3
12. 12.What is the remainder when 188 is divided by 7? (a) 6 (b) 1 (c) 4 (d) 5
13. 13.What is the remainder when 115 is divided by 6? (a) 5 (b) 2 (c) 0 (d) 1
14. 14.What is the smallest number you must add to 199 to make it divisible by 6? (a) 3 (b) 4 (c) 1 (d) 5
15. 15.What is the remainder when 205 is divided by 3? (a) 1 (b) 2 (c) 0
16. 16.What is the units digit of 3^19? (a) 9 (b) 3 (c) 7 (d) 1
17. 17.Which of these numbers is divisible by 3? (a) 1787 (b) 1793 (c) 1788 (d) 1789
18. 18.What is the remainder when 604 is divided by 10? (a) 4 (b) 6 (c) 3 (d) 5
19. 19.What is the smallest number you must add to 235 to make it divisible by 6? (a) 3 (b) 5 (c) 1 (d) 4
20. 20.What is the units digit of 2^54? (a) 8 (b) 2 (c) 6 (d) 4
21. 21.Which of these numbers is divisible by 3? (a) 367 (b) 365 (c) 362 (d) 363
22. 22.What is the units digit of 7^60? (a) 3 (b) 1 (c) 9 (d) 7
23. 23.Which of these numbers is divisible by 9? (a) 1075 (b) 1061 (c) 1062 (d) 1063
24. 24.Which of these numbers is divisible by 4? (a) 2211 (b) 2205 (c) 2208 (d) 2213
25. 25.Which of these numbers is divisible by 8? (a) 4749 (b) 4759 (c) 4744 (d) 4743
26. 26.Which of these numbers is divisible by 8? (a) 2424 (b) 2436 (c) 2422 (d) 2429
27. 27.What is the remainder when 831 is divided by 12? (a) 9 (b) 2 (c) 4 (d) 3
28. 28.What is the smallest number you must add to 103 to make it divisible by 3? (a) 2 (b) 1
29. 29.Which of these numbers is divisible by 11? (a) 8268 (b) 8280 (c) 8261 (d) 8259
30. 30.Which of these numbers is divisible by 6? (a) 1628 (b) 1633 (c) 1641 (d) 1632
31. 31.What is the smallest number you must add to 409 to make it divisible by 7? (a) 3 (b) 6 (c) 4 (d) 5
32. 32.What is the remainder when 535 is divided by 10? (a) 5 (b) 7 (c) 6 (d) 4
33. 33.What is the units digit of 3^32? (a) 9 (b) 7 (c) 1 (d) 3
34. 34.What is the smallest number you must add to 226 to make it divisible by 3? (a) 2 (b) 1
35. 35.What is the remainder when 714 is divided by 11? (a) 1 (b) 8 (c) 9 (d) 10
36. 36.Which of these numbers is divisible by 9? (a) 5976 (b) 5988 (c) 5969 (d) 5977
37. 37.What is the remainder when 485 is divided by 6? (a) 5 (b) 3 (c) 4 (d) 1
38. 38.What is the remainder when 209 is divided by 12? (a) 6 (b) 5 (c) 7 (d) 4
39. 39.What is the remainder when 596 is divided by 9? (a) 1 (b) 2 (c) 7 (d) 3
40. 40.Which of these numbers is divisible by 9? (a) 3589 (b) 3568 (c) 3573 (d) 3577

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