Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 702 · 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 344 is divided by 9? (a) 3 (b) 7 (c) 2 (d) 1
2. 2.What is the units digit of 7^13? (a) 9 (b) 7 (c) 1 (d) 3
3. 3.Which of these numbers is divisible by 9? (a) 1669 (b) 1649 (c) 1662 (d) 1656
4. 4.What is the smallest number you must add to 177 to make it divisible by 5? (a) 3 (b) 2 (c) 1 (d) 4
5. 5.What is the remainder when 326 is divided by 12? (a) 2 (b) 10 (c) 3 (d) 1
6. 6.Which of these numbers is divisible by 9? (a) 682 (b) 675 (c) 669 (d) 687
7. 7.What is the units digit of 4^30? (a) 7 (b) 4 (c) 6 (d) 5
8. 8.What is the units digit of 3^59? (a) 3 (b) 7 (c) 9 (d) 1
9. 9.What is the units digit of 9^14? (a) 9 (b) 2 (c) 0 (d) 1
10. 10.What is the units digit of 2^49? (a) 2 (b) 4 (c) 8 (d) 6
11. 11.What is the remainder when 315 is divided by 6? (a) 4 (b) 2 (c) 3 (d) 5
12. 12.What is the smallest number you must add to 367 to make it divisible by 9? (a) 3 (b) 1 (c) 2 (d) 7
13. 13.What is the remainder when 87 is divided by 4? (a) 2 (b) 0 (c) 1 (d) 3
14. 14.What is the smallest number you must add to 85 to make it divisible by 8? (a) 4 (b) 3 (c) 2 (d) 5
15. 15.What is the smallest number you must add to 186 to make it divisible by 12? (a) 6 (b) 5 (c) 8 (d) 7
16. 16.What is the smallest number you must add to 199 to make it divisible by 4? (a) 3 (b) 1 (c) 2
17. 17.What is the smallest number you must add to 219 to make it divisible by 11? (a) 3 (b) 10 (c) 1 (d) 2
18. 18.What is the remainder when 796 is divided by 12? (a) 5 (b) 3 (c) 8 (d) 4
19. 19.Which of these numbers is divisible by 3? (a) 2581 (b) 2583 (c) 2588 (d) 2585
20. 20.What is the units digit of 7^18? (a) 1 (b) 9 (c) 3 (d) 7
21. 21.What is the smallest number you must add to 122 to make it divisible by 4? (a) 1 (b) 3 (c) 2
22. 22.What is the remainder when 379 is divided by 5? (a) 2 (b) 4 (c) 1 (d) 3
23. 23.Which of these numbers is divisible by 9? (a) 7722 (b) 7719 (c) 7736 (d) 7727
24. 24.What is the remainder when 173 is divided by 3? (a) 2 (b) 0 (c) 1
25. 25.What is the remainder when 545 is divided by 8? (a) 2 (b) 7 (c) 0 (d) 1
26. 26.What is the remainder when 318 is divided by 8? (a) 6 (b) 7 (c) 2 (d) 5
27. 27.Which of these numbers is divisible by 8? (a) 1098 (b) 1092 (c) 1107 (d) 1096
28. 28.What is the units digit of 9^34? (a) 1 (b) 0 (c) 2 (d) 9
29. 29.Which of these numbers is divisible by 3? (a) 2165 (b) 2166 (c) 2168 (d) 2170
30. 30.Which of these numbers is divisible by 6? (a) 2086 (b) 2093 (c) 2081 (d) 2082
31. 31.What is the smallest number you must add to 102 to make it divisible by 10? (a) 7 (b) 2 (c) 9 (d) 8
32. 32.What is the units digit of 2^31? (a) 4 (b) 8 (c) 6 (d) 2
33. 33.What is the units digit of 4^50? (a) 7 (b) 5 (c) 6 (d) 4
34. 34.What is the units digit of 8^15? (a) 2 (b) 8 (c) 6 (d) 4
35. 35.What is the units digit of 2^34? (a) 8 (b) 6 (c) 2 (d) 4
36. 36.What is the units digit of 2^42? (a) 8 (b) 4 (c) 2 (d) 6
37. 37.Which of these numbers is divisible by 8? (a) 5657 (b) 5668 (c) 5656 (d) 5653
38. 38.What is the units digit of 2^18? (a) 6 (b) 8 (c) 2 (d) 4
39. 39.What is the units digit of 4^25? (a) 4 (b) 6 (c) 3 (d) 5
40. 40.What is the remainder when 269 is divided by 4? (a) 2 (b) 0 (c) 3 (d) 1

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