Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 278 · 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 645 is divided by 8? (a) 4 (b) 3 (c) 5 (d) 6
2. 2.What is the remainder when 411 is divided by 7? (a) 4 (b) 6 (c) 5 (d) 2
3. 3.What is the units digit of 4^16? (a) 4 (b) 6 (c) 7 (d) 5
4. 4.What is the remainder when 157 is divided by 11? (a) 8 (b) 2 (c) 3 (d) 4
5. 5.What is the smallest number you must add to 666 to make it divisible by 11? (a) 6 (b) 5 (c) 4 (d) 7
6. 6.Which of these numbers is divisible by 4? (a) 2162 (b) 2158 (c) 2160 (d) 2167
7. 7.What is the smallest number you must add to 149 to make it divisible by 3? (a) 2 (b) 1
8. 8.What is the units digit of 7^11? (a) 9 (b) 3 (c) 7 (d) 1
9. 9.Which of these numbers is divisible by 8? (a) 1107 (b) 1096 (c) 1097 (d) 1093
10. 10.What is the units digit of 7^40? (a) 7 (b) 3 (c) 9 (d) 1
11. 11.What is the units digit of 2^57? (a) 8 (b) 6 (c) 4 (d) 2
12. 12.What is the remainder when 116 is divided by 9? (a) 7 (b) 6 (c) 8 (d) 1
13. 13.What is the units digit of 4^26? (a) 4 (b) 6 (c) 7 (d) 5
14. 14.What is the remainder when 267 is divided by 4? (a) 0 (b) 1 (c) 3 (d) 2
15. 15.What is the units digit of 2^36? (a) 4 (b) 2 (c) 6 (d) 8
16. 16.What is the remainder when 239 is divided by 12? (a) 11 (b) 10 (c) 9 (d) 1
17. 17.What is the remainder when 177 is divided by 12? (a) 8 (b) 10 (c) 3 (d) 9
18. 18.What is the units digit of 2^20? (a) 8 (b) 2 (c) 4 (d) 6
19. 19.What is the units digit of 3^33? (a) 7 (b) 9 (c) 1 (d) 3
20. 20.Which of these numbers is divisible by 9? (a) 4025 (b) 4021 (c) 4040 (d) 4023
21. 21.What is the smallest number you must add to 633 to make it divisible by 8? (a) 6 (b) 5 (c) 7 (d) 1
22. 22.What is the units digit of 7^49? (a) 7 (b) 3 (c) 9 (d) 1
23. 23.What is the remainder when 263 is divided by 6? (a) 3 (b) 4 (c) 5 (d) 1
24. 24.Which of these numbers is divisible by 6? (a) 3891 (b) 3901 (c) 3894 (d) 3898
25. 25.What is the remainder when 71 is divided by 3? (a) 0 (b) 1 (c) 2
26. 26.Which of these numbers is divisible by 6? (a) 4877 (b) 4868 (c) 4866 (d) 4862
27. 27.What is the remainder when 391 is divided by 5? (a) 1 (b) 0 (c) 4 (d) 2
28. 28.What is the units digit of 7^28? (a) 1 (b) 9 (c) 7 (d) 3
29. 29.Which of these numbers is divisible by 8? (a) 1179 (b) 1171 (c) 1176 (d) 1189
30. 30.What is the smallest number you must add to 485 to make it divisible by 7? (a) 6 (b) 4 (c) 2 (d) 5
31. 31.What is the smallest number you must add to 313 to make it divisible by 5? (a) 4 (b) 3 (c) 2 (d) 1
32. 32.What is the units digit of 8^50? (a) 6 (b) 8 (c) 2 (d) 4
33. 33.Which of these numbers is divisible by 8? (a) 5684 (b) 5672 (c) 5679 (d) 5671
34. 34.Which of these numbers is divisible by 6? (a) 3386 (b) 3394 (c) 3397 (d) 3390
35. 35.What is the units digit of 3^53? (a) 1 (b) 9 (c) 7 (d) 3
36. 36.What is the units digit of 7^15? (a) 3 (b) 7 (c) 1 (d) 9
37. 37.What is the smallest number you must add to 857 to make it divisible by 12? (a) 8 (b) 6 (c) 5 (d) 7
38. 38.What is the units digit of 2^10? (a) 6 (b) 8 (c) 2 (d) 4
39. 39.What is the units digit of 2^49? (a) 6 (b) 2 (c) 4 (d) 8
40. 40.What is the units digit of 2^23? (a) 2 (b) 8 (c) 6 (d) 4

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