Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 899 · 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 481 is divided by 6? (a) 5 (b) 2 (c) 0 (d) 1
2. 2.What is the units digit of 7^36? (a) 9 (b) 7 (c) 3 (d) 1
3. 3.What is the remainder when 929 is divided by 12? (a) 4 (b) 6 (c) 7 (d) 5
4. 4.What is the units digit of 7^12? (a) 9 (b) 3 (c) 7 (d) 1
5. 5.What is the smallest number you must add to 337 to make it divisible by 8? (a) 7 (b) 6 (c) 1 (d) 5
6. 6.What is the units digit of 9^13? (a) 1 (b) 9 (c) 7 (d) 8
7. 7.Which of these numbers is divisible by 11? (a) 3846 (b) 3835 (c) 3818 (d) 3828
8. 8.What is the smallest number you must add to 113 to make it divisible by 4? (a) 1 (b) 2 (c) 3
9. 9.Which of these numbers is divisible by 3? (a) 612 (b) 616 (c) 613 (d) 610
10. 10.What is the remainder when 169 is divided by 3? (a) 2 (b) 1 (c) 0
11. 11.Which of these numbers is divisible by 8? (a) 4092 (b) 4086 (c) 4088 (d) 4098
12. 12.What is the units digit of 2^12? (a) 4 (b) 6 (c) 2 (d) 8
13. 13.What is the remainder when 221 is divided by 5? (a) 2 (b) 1 (c) 0 (d) 4
14. 14.Which of these numbers is divisible by 6? (a) 3649 (b) 3654 (c) 3665 (d) 3656
15. 15.Which of these numbers is divisible by 9? (a) 5899 (b) 5886 (c) 5885 (d) 5891
16. 16.What is the smallest number you must add to 205 to make it divisible by 3? (a) 2 (b) 1
17. 17.What is the units digit of 8^17? (a) 4 (b) 8 (c) 2 (d) 6
18. 18.Which of these numbers is divisible by 8? (a) 472 (b) 476 (c) 467 (d) 486
19. 19.What is the remainder when 602 is divided by 10? (a) 8 (b) 3 (c) 2 (d) 1
20. 20.What is the remainder when 184 is divided by 3? (a) 0 (b) 1 (c) 2
21. 21.Which of these numbers is divisible by 6? (a) 1416 (b) 1426 (c) 1414 (d) 1421
22. 22.What is the remainder when 242 is divided by 3? (a) 1 (b) 2 (c) 0
23. 23.Which of these numbers is divisible by 8? (a) 3090 (b) 3111 (c) 3096 (d) 3097
24. 24.Which of these numbers is divisible by 9? (a) 7908 (b) 7923 (c) 7915 (d) 7911
25. 25.Which of these numbers is divisible by 6? (a) 4444 (b) 4438 (c) 4447 (d) 4440
26. 26.What is the remainder when 152 is divided by 10? (a) 3 (b) 2 (c) 8 (d) 1
27. 27.What is the remainder when 268 is divided by 7? (a) 5 (b) 1 (c) 3 (d) 2
28. 28.What is the smallest number you must add to 124 to make it divisible by 5? (a) 1 (b) 2 (c) 4 (d) 3
29. 29.What is the remainder when 95 is divided by 9? (a) 7 (b) 6 (c) 5 (d) 4
30. 30.What is the remainder when 645 is divided by 8? (a) 4 (b) 6 (c) 5 (d) 3
31. 31.What is the units digit of 4^55? (a) 5 (b) 6 (c) 3 (d) 4
32. 32.What is the smallest number you must add to 356 to make it divisible by 5? (a) 2 (b) 1 (c) 4 (d) 3
33. 33.What is the smallest number you must add to 71 to make it divisible by 7? (a) 4 (b) 5 (c) 1 (d) 6
34. 34.Which of these numbers is divisible by 6? (a) 1961 (b) 1969 (c) 1967 (d) 1962
35. 35.What is the smallest number you must add to 289 to make it divisible by 5? (a) 3 (b) 2 (c) 1 (d) 4
36. 36.What is the units digit of 3^34? (a) 7 (b) 9 (c) 1 (d) 3
37. 37.What is the smallest number you must add to 210 to make it divisible by 11? (a) 1 (b) 8 (c) 9 (d) 10
38. 38.Which of these numbers is divisible by 11? (a) 4774 (b) 4772 (c) 4787 (d) 4781
39. 39.Which of these numbers is divisible by 4? (a) 234 (b) 226 (c) 228 (d) 231
40. 40.What is the units digit of 3^15? (a) 3 (b) 1 (c) 9 (d) 7

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