Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 683 · 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 8^28? (a) 6 (b) 8 (c) 2 (d) 4
2. 2.What is the remainder when 179 is divided by 5? (a) 4 (b) 1 (c) 3 (d) 2
3. 3.Which of these numbers is divisible by 3? (a) 976 (b) 975 (c) 973 (d) 980
4. 4.What is the units digit of 9^54? (a) 0 (b) 1 (c) 9 (d) 2
5. 5.What is the smallest number you must add to 104 to make it divisible by 10? (a) 5 (b) 4 (c) 6 (d) 7
6. 6.What is the smallest number you must add to 93 to make it divisible by 5? (a) 2 (b) 1 (c) 4 (d) 3
7. 7.What is the smallest number you must add to 197 to make it divisible by 12? (a) 6 (b) 7 (c) 8 (d) 5
8. 8.Which of these numbers is divisible by 8? (a) 6677 (b) 6667 (c) 6672 (d) 6682
9. 9.What is the remainder when 344 is divided by 11? (a) 4 (b) 2 (c) 3 (d) 8
10. 10.What is the units digit of 3^44? (a) 9 (b) 7 (c) 3 (d) 1
11. 11.What is the smallest number you must add to 582 to make it divisible by 12? (a) 8 (b) 7 (c) 6 (d) 5
12. 12.What is the smallest number you must add to 382 to make it divisible by 8? (a) 2 (b) 3 (c) 6 (d) 1
13. 13.What is the remainder when 98 is divided by 3? (a) 2 (b) 1 (c) 0
14. 14.Which of these numbers is divisible by 9? (a) 4131 (b) 4128 (c) 4148 (d) 4138
15. 15.Which of these numbers is divisible by 11? (a) 7926 (b) 7920 (c) 7910 (d) 7935
16. 16.What is the remainder when 166 is divided by 8? (a) 2 (b) 6 (c) 5 (d) 7
17. 17.Which of these numbers is divisible by 9? (a) 5941 (b) 5923 (c) 5938 (d) 5931
18. 18.Which of these numbers is divisible by 6? (a) 731 (b) 719 (c) 720 (d) 722
19. 19.What is the remainder when 91 is divided by 4? (a) 2 (b) 1 (c) 0 (d) 3
20. 20.Which of these numbers is divisible by 6? (a) 664 (b) 660 (c) 659 (d) 671
21. 21.What is the smallest number you must add to 57 to make it divisible by 5? (a) 2 (b) 4 (c) 1 (d) 3
22. 22.What is the remainder when 349 is divided by 10? (a) 8 (b) 1 (c) 7 (d) 9
23. 23.What is the smallest number you must add to 938 to make it divisible by 12? (a) 10 (b) 11 (c) 9 (d) 2
24. 24.Which of these numbers is divisible by 8? (a) 760 (b) 762 (c) 753 (d) 773
25. 25.Which of these numbers is divisible by 11? (a) 1039 (b) 1024 (c) 1022 (d) 1023
26. 26.What is the remainder when 257 is divided by 4? (a) 3 (b) 2 (c) 0 (d) 1
27. 27.Which of these numbers is divisible by 8? (a) 598 (b) 584 (c) 587 (d) 583
28. 28.What is the units digit of 8^17? (a) 8 (b) 2 (c) 6 (d) 4
29. 29.Which of these numbers is divisible by 8? (a) 7058 (b) 7052 (c) 7046 (d) 7048
30. 30.What is the smallest number you must add to 262 to make it divisible by 5? (a) 1 (b) 2 (c) 3 (d) 4
31. 31.Which of these numbers is divisible by 3? (a) 1508 (b) 1511 (c) 1514 (d) 1509
32. 32.What is the smallest number you must add to 334 to make it divisible by 6? (a) 3 (b) 4 (c) 2 (d) 1
33. 33.What is the smallest number you must add to 219 to make it divisible by 6? (a) 4 (b) 5 (c) 2 (d) 3
34. 34.What is the units digit of 2^16? (a) 6 (b) 4 (c) 8 (d) 2
35. 35.Which of these numbers is divisible by 4? (a) 1154 (b) 1151 (c) 1147 (d) 1148
36. 36.What is the units digit of 3^30? (a) 3 (b) 9 (c) 7 (d) 1
37. 37.What is the units digit of 3^59? (a) 7 (b) 3 (c) 9 (d) 1
38. 38.What is the smallest number you must add to 307 to make it divisible by 7? (a) 6 (b) 1 (c) 3 (d) 2
39. 39.What is the remainder when 59 is divided by 3? (a) 1 (b) 0 (c) 2
40. 40.What is the smallest number you must add to 239 to make it divisible by 4? (a) 1 (b) 3 (c) 2

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