Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 495 · 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 2^18? (a) 2 (b) 6 (c) 8 (d) 4
2. 2.What is the smallest number you must add to 211 to make it divisible by 3? (a) 1 (b) 2
3. 3.What is the units digit of 3^16? (a) 9 (b) 7 (c) 1 (d) 3
4. 4.What is the units digit of 7^22? (a) 1 (b) 7 (c) 9 (d) 3
5. 5.Which of these numbers is divisible by 8? (a) 2523 (b) 2514 (c) 2512 (d) 2507
6. 6.What is the units digit of 4^46? (a) 7 (b) 5 (c) 6 (d) 4
7. 7.What is the remainder when 541 is divided by 8? (a) 6 (b) 5 (c) 3 (d) 4
8. 8.What is the smallest number you must add to 544 to make it divisible by 7? (a) 3 (b) 1 (c) 5 (d) 2
9. 9.Which of these numbers is divisible by 3? (a) 1661 (b) 1663 (c) 1667 (d) 1662
10. 10.What is the remainder when 435 is divided by 6? (a) 4 (b) 2 (c) 5 (d) 3
11. 11.What is the remainder when 58 is divided by 4? (a) 1 (b) 0 (c) 3 (d) 2
12. 12.What is the remainder when 362 is divided by 7? (a) 2 (b) 6 (c) 5 (d) 4
13. 13.What is the units digit of 7^15? (a) 1 (b) 9 (c) 3 (d) 7
14. 14.What is the smallest number you must add to 173 to make it divisible by 3? (a) 2 (b) 1
15. 15.What is the remainder when 231 is divided by 4? (a) 1 (b) 2 (c) 3 (d) 0
16. 16.What is the smallest number you must add to 345 to make it divisible by 8? (a) 6 (b) 1 (c) 5 (d) 7
17. 17.Which of these numbers is divisible by 6? (a) 466 (b) 473 (c) 462 (d) 460
18. 18.Which of these numbers is divisible by 3? (a) 2083 (b) 2078 (c) 2079 (d) 2080
19. 19.What is the remainder when 384 is divided by 5? (a) 3 (b) 2 (c) 4 (d) 1
20. 20.What is the smallest number you must add to 333 to make it divisible by 6? (a) 2 (b) 3 (c) 4 (d) 5
21. 21.What is the remainder when 688 is divided by 11? (a) 8 (b) 7 (c) 6 (d) 5
22. 22.Which of these numbers is divisible by 4? (a) 1648 (b) 1654 (c) 1647 (d) 1651
23. 23.What is the remainder when 121 is divided by 6? (a) 5 (b) 2 (c) 0 (d) 1
24. 24.What is the smallest number you must add to 861 to make it divisible by 12? (a) 2 (b) 3 (c) 4 (d) 9
25. 25.Which of these numbers is divisible by 11? (a) 8261 (b) 8277 (c) 8251 (d) 8271
26. 26.What is the remainder when 319 is divided by 6? (a) 2 (b) 0 (c) 5 (d) 1
27. 27.What is the remainder when 419 is divided by 10? (a) 1 (b) 7 (c) 9 (d) 8
28. 28.Which of these numbers is divisible by 6? (a) 3935 (b) 3923 (c) 3924 (d) 3925
29. 29.What is the smallest number you must add to 475 to make it divisible by 8? (a) 5 (b) 4 (c) 6 (d) 3
30. 30.What is the smallest number you must add to 214 to make it divisible by 6? (a) 4 (b) 3 (c) 1 (d) 2
31. 31.What is the smallest number you must add to 50 to make it divisible by 3? (a) 1 (b) 2
32. 32.What is the smallest number you must add to 307 to make it divisible by 4? (a) 3 (b) 2 (c) 1
33. 33.What is the remainder when 369 is divided by 6? (a) 2 (b) 3 (c) 4 (d) 5
34. 34.What is the units digit of 8^15? (a) 6 (b) 4 (c) 2 (d) 8
35. 35.Which of these numbers is divisible by 3? (a) 1563 (b) 1561 (c) 1564 (d) 1567
36. 36.Which of these numbers is divisible by 3? (a) 1843 (b) 1842 (c) 1840 (d) 1847
37. 37.Which of these numbers is divisible by 4? (a) 2030 (b) 2027 (c) 2024 (d) 2022
38. 38.What is the smallest number you must add to 54 to make it divisible by 5? (a) 4 (b) 2 (c) 1 (d) 3
39. 39.What is the remainder when 263 is divided by 4? (a) 3 (b) 2 (c) 1 (d) 0
40. 40.What is the remainder when 295 is divided by 4? (a) 1 (b) 3 (c) 0 (d) 2

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