Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 871 · 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 7^39? (a) 7 (b) 1 (c) 9 (d) 3
2. 2.What is the units digit of 9^47? (a) 8 (b) 9 (c) 1 (d) 7
3. 3.What is the remainder when 803 is divided by 10? (a) 7 (b) 4 (c) 2 (d) 3
4. 4.What is the remainder when 663 is divided by 11? (a) 3 (b) 8 (c) 4 (d) 2
5. 5.Which of these numbers is divisible by 8? (a) 7191 (b) 7182 (c) 7175 (d) 7176
6. 6.What is the remainder when 226 is divided by 7? (a) 2 (b) 1 (c) 3 (d) 5
7. 7.What is the units digit of 3^36? (a) 1 (b) 3 (c) 9 (d) 7
8. 8.Which of these numbers is divisible by 4? (a) 268 (b) 269 (c) 274 (d) 266
9. 9.What is the units digit of 2^22? (a) 4 (b) 2 (c) 6 (d) 8
10. 10.Which of these numbers is divisible by 4? (a) 1453 (b) 1448 (c) 1450 (d) 1446
11. 11.What is the smallest number you must add to 370 to make it divisible by 9? (a) 8 (b) 7 (c) 1 (d) 6
12. 12.What is the remainder when 112 is divided by 3? (a) 0 (b) 2 (c) 1
13. 13.Which of these numbers is divisible by 3? (a) 514 (b) 520 (c) 516 (d) 518
14. 14.Which of these numbers is divisible by 8? (a) 5080 (b) 5081 (c) 5073 (d) 5091
15. 15.What is the units digit of 7^27? (a) 1 (b) 3 (c) 7 (d) 9
16. 16.What is the smallest number you must add to 204 to make it divisible by 8? (a) 6 (b) 3 (c) 4 (d) 5
17. 17.What is the remainder when 106 is divided by 5? (a) 0 (b) 1 (c) 2 (d) 4
18. 18.What is the remainder when 512 is divided by 7? (a) 6 (b) 2 (c) 0 (d) 1
19. 19.What is the smallest number you must add to 196 to make it divisible by 8? (a) 3 (b) 4 (c) 5 (d) 6
20. 20.What is the smallest number you must add to 283 to make it divisible by 7? (a) 6 (b) 3 (c) 4 (d) 5
21. 21.What is the remainder when 304 is divided by 6? (a) 2 (b) 3 (c) 4 (d) 5
22. 22.Which of these numbers is divisible by 4? (a) 1930 (b) 1925 (c) 1928 (d) 1934
23. 23.What is the smallest number you must add to 128 to make it divisible by 3? (a) 2 (b) 1
24. 24.What is the units digit of 3^44? (a) 9 (b) 3 (c) 1 (d) 7
25. 25.What is the units digit of 3^10? (a) 7 (b) 1 (c) 3 (d) 9
26. 26.Which of these numbers is divisible by 3? (a) 1480 (b) 1484 (c) 1487 (d) 1482
27. 27.Which of these numbers is divisible by 3? (a) 1776 (b) 1781 (c) 1777 (d) 1775
28. 28.Which of these numbers is divisible by 4? (a) 1588 (b) 1593 (c) 1587 (d) 1589
29. 29.What is the remainder when 148 is divided by 5? (a) 1 (b) 3 (c) 4 (d) 2
30. 30.What is the smallest number you must add to 226 to make it divisible by 12? (a) 10 (b) 3 (c) 1 (d) 2
31. 31.What is the units digit of 3^20? (a) 7 (b) 9 (c) 1 (d) 3
32. 32.Which of these numbers is divisible by 9? (a) 7519 (b) 7515 (c) 7514 (d) 7532
33. 33.What is the remainder when 221 is divided by 12? (a) 5 (b) 4 (c) 6 (d) 7
34. 34.Which of these numbers is divisible by 4? (a) 2450 (b) 2458 (c) 2455 (d) 2452
35. 35.What is the remainder when 66 is divided by 4? (a) 0 (b) 2 (c) 3 (d) 1
36. 36.What is the remainder when 40 is divided by 3? (a) 0 (b) 1 (c) 2
37. 37.What is the units digit of 4^32? (a) 4 (b) 6 (c) 7 (d) 5
38. 38.What is the remainder when 382 is divided by 8? (a) 7 (b) 5 (c) 6 (d) 2
39. 39.What is the units digit of 2^19? (a) 4 (b) 6 (c) 2 (d) 8
40. 40.What is the units digit of 4^16? (a) 6 (b) 4 (c) 7 (d) 5

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