Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 348 · 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.Which of these numbers is divisible by 6? (a) 2194 (b) 2207 (c) 2196 (d) 2197
2. 2.Which of these numbers is divisible by 8? (a) 4904 (b) 4917 (c) 4909 (d) 4897
3. 3.What is the smallest number you must add to 272 to make it divisible by 5? (a) 4 (b) 2 (c) 3 (d) 1
4. 4.Which of these numbers is divisible by 4? (a) 3560 (b) 3559 (c) 3561 (d) 3565
5. 5.Which of these numbers is divisible by 11? (a) 7764 (b) 7744 (c) 7740 (d) 7750
6. 6.What is the smallest number you must add to 130 to make it divisible by 3? (a) 2 (b) 1
7. 7.What is the remainder when 575 is divided by 12? (a) 11 (b) 10 (c) 9 (d) 1
8. 8.What is the remainder when 164 is divided by 5? (a) 2 (b) 1 (c) 4 (d) 3
9. 9.Which of these numbers is divisible by 6? (a) 4865 (b) 4855 (c) 4870 (d) 4860
10. 10.What is the units digit of 2^19? (a) 6 (b) 4 (c) 2 (d) 8
11. 11.Which of these numbers is divisible by 4? (a) 917 (b) 916 (c) 914 (d) 922
12. 12.What is the units digit of 4^43? (a) 5 (b) 6 (c) 3 (d) 4
13. 13.Which of these numbers is divisible by 8? (a) 6957 (b) 6952 (c) 6947 (d) 6963
14. 14.What is the units digit of 7^34? (a) 3 (b) 9 (c) 7 (d) 1
15. 15.What is the remainder when 185 is divided by 4? (a) 3 (b) 1 (c) 0 (d) 2
16. 16.What is the remainder when 124 is divided by 3? (a) 2 (b) 0 (c) 1
17. 17.What is the remainder when 134 is divided by 4? (a) 3 (b) 1 (c) 2 (d) 0
18. 18.What is the units digit of 3^27? (a) 9 (b) 7 (c) 3 (d) 1
19. 19.What is the units digit of 9^50? (a) 2 (b) 9 (c) 0 (d) 1
20. 20.Which of these numbers is divisible by 4? (a) 3270 (b) 3268 (c) 3273 (d) 3267
21. 21.What is the smallest number you must add to 826 to make it divisible by 12? (a) 2 (b) 10 (c) 1 (d) 3
22. 22.Which of these numbers is divisible by 8? (a) 4993 (b) 4988 (c) 4992 (d) 5002
23. 23.Which of these numbers is divisible by 8? (a) 3969 (b) 3965 (c) 3983 (d) 3968
24. 24.What is the smallest number you must add to 427 to make it divisible by 11? (a) 3 (b) 1 (c) 2 (d) 9
25. 25.What is the smallest number you must add to 172 to make it divisible by 5? (a) 1 (b) 2 (c) 3 (d) 4
26. 26.Which of these numbers is divisible by 11? (a) 5410 (b) 5417 (c) 5394 (d) 5401
27. 27.What is the units digit of 9^55? (a) 7 (b) 8 (c) 9 (d) 1
28. 28.Which of these numbers is divisible by 11? (a) 5193 (b) 5203 (c) 5209 (d) 5215
29. 29.What is the units digit of 3^17? (a) 7 (b) 1 (c) 9 (d) 3
30. 30.What is the units digit of 4^54? (a) 7 (b) 5 (c) 4 (d) 6
31. 31.What is the units digit of 9^38? (a) 2 (b) 9 (c) 1 (d) 0
32. 32.What is the smallest number you must add to 381 to make it divisible by 12? (a) 4 (b) 9 (c) 3 (d) 2
33. 33.What is the units digit of 3^12? (a) 3 (b) 7 (c) 1 (d) 9
34. 34.Which of these numbers is divisible by 6? (a) 916 (b) 918 (c) 925 (d) 920
35. 35.What is the units digit of 9^36? (a) 0 (b) 9 (c) 2 (d) 1
36. 36.What is the units digit of 8^34? (a) 6 (b) 2 (c) 4 (d) 8
37. 37.Which of these numbers is divisible by 4? (a) 2676 (b) 2674 (c) 2681 (d) 2677
38. 38.What is the remainder when 65 is divided by 3? (a) 2 (b) 1 (c) 0
39. 39.Which of these numbers is divisible by 8? (a) 3845 (b) 3849 (c) 3857 (d) 3848
40. 40.Which of these numbers is divisible by 8? (a) 3287 (b) 3280 (c) 3278 (d) 3294

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