Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 858 · 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 3^21? (a) 9 (b) 1 (c) 3 (d) 7
2. 2.What is the units digit of 9^57? (a) 1 (b) 7 (c) 8 (d) 9
3. 3.What is the remainder when 153 is divided by 5? (a) 1 (b) 4 (c) 3 (d) 2
4. 4.What is the remainder when 250 is divided by 4? (a) 2 (b) 1 (c) 0 (d) 3
5. 5.Which of these numbers is divisible by 4? (a) 1385 (b) 1379 (c) 1381 (d) 1380
6. 6.Which of these numbers is divisible by 3? (a) 1317 (b) 1321 (c) 1318 (d) 1315
7. 7.What is the remainder when 178 is divided by 9? (a) 6 (b) 8 (c) 7 (d) 2
8. 8.What is the remainder when 70 is divided by 3? (a) 0 (b) 1 (c) 2
9. 9.What is the units digit of 8^13? (a) 4 (b) 6 (c) 8 (d) 2
10. 10.What is the units digit of 7^10? (a) 9 (b) 1 (c) 3 (d) 7
11. 11.What is the smallest number you must add to 298 to make it divisible by 11? (a) 9 (b) 1 (c) 8 (d) 10
12. 12.What is the units digit of 8^25? (a) 8 (b) 2 (c) 6 (d) 4
13. 13.What is the smallest number you must add to 158 to make it divisible by 12? (a) 2 (b) 9 (c) 11 (d) 10
14. 14.What is the smallest number you must add to 212 to make it divisible by 5? (a) 4 (b) 2 (c) 1 (d) 3
15. 15.What is the smallest number you must add to 386 to make it divisible by 6? (a) 2 (b) 5 (c) 3 (d) 4
16. 16.What is the units digit of 9^20? (a) 9 (b) 2 (c) 0 (d) 1
17. 17.Which of these numbers is divisible by 9? (a) 3308 (b) 3299 (c) 3316 (d) 3303
18. 18.What is the smallest number you must add to 335 to make it divisible by 12? (a) 11 (b) 1 (c) 3 (d) 2
19. 19.What is the units digit of 7^58? (a) 7 (b) 3 (c) 9 (d) 1
20. 20.What is the smallest number you must add to 157 to make it divisible by 9? (a) 7 (b) 5 (c) 6 (d) 4
21. 21.What is the remainder when 477 is divided by 11? (a) 5 (b) 7 (c) 4 (d) 3
22. 22.What is the units digit of 9^36? (a) 0 (b) 2 (c) 9 (d) 1
23. 23.What is the units digit of 2^51? (a) 6 (b) 4 (c) 2 (d) 8
24. 24.What is the remainder when 465 is divided by 10? (a) 7 (b) 5 (c) 6 (d) 4
25. 25.What is the remainder when 103 is divided by 5? (a) 1 (b) 4 (c) 3 (d) 2
26. 26.Which of these numbers is divisible by 6? (a) 861 (b) 867 (c) 871 (d) 864
27. 27.What is the smallest number you must add to 687 to make it divisible by 10? (a) 3 (b) 7 (c) 4 (d) 2
28. 28.What is the smallest number you must add to 466 to make it divisible by 8? (a) 6 (b) 7 (c) 2 (d) 5
29. 29.What is the smallest number you must add to 117 to make it divisible by 7? (a) 3 (b) 1 (c) 2 (d) 5
30. 30.What is the smallest number you must add to 179 to make it divisible by 5? (a) 1 (b) 3 (c) 2 (d) 4
31. 31.What is the remainder when 82 is divided by 4? (a) 0 (b) 2 (c) 3 (d) 1
32. 32.Which of these numbers is divisible by 11? (a) 4771 (b) 4784 (c) 4763 (d) 4762
33. 33.Which of these numbers is divisible by 8? (a) 3328 (b) 3341 (c) 3334 (d) 3324
34. 34.What is the smallest number you must add to 70 to make it divisible by 4? (a) 1 (b) 2 (c) 3
35. 35.What is the units digit of 9^42? (a) 9 (b) 2 (c) 1 (d) 0
36. 36.What is the units digit of 9^24? (a) 0 (b) 9 (c) 2 (d) 1
37. 37.What is the remainder when 603 is divided by 8? (a) 4 (b) 3 (c) 2 (d) 5
38. 38.What is the units digit of 2^26? (a) 4 (b) 6 (c) 2 (d) 8
39. 39.What is the units digit of 7^18? (a) 3 (b) 9 (c) 1 (d) 7
40. 40.What is the smallest number you must add to 168 to make it divisible by 11? (a) 9 (b) 7 (c) 8 (d) 3

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