Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 246 · 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 8? (a) 6679 (b) 6691 (c) 6680 (d) 6686
2. 2.Which of these numbers is divisible by 9? (a) 1192 (b) 1179 (c) 1173 (d) 1185
3. 3.What is the units digit of 7^36? (a) 7 (b) 9 (c) 3 (d) 1
4. 4.What is the remainder when 78 is divided by 5? (a) 1 (b) 2 (c) 4 (d) 3
5. 5.What is the units digit of 3^20? (a) 7 (b) 1 (c) 3 (d) 9
6. 6.What is the remainder when 778 is divided by 11? (a) 3 (b) 8 (c) 9 (d) 7
7. 7.What is the smallest number you must add to 522 to make it divisible by 12? (a) 8 (b) 5 (c) 6 (d) 7
8. 8.What is the smallest number you must add to 171 to make it divisible by 5? (a) 2 (b) 3 (c) 1 (d) 4
9. 9.Which of these numbers is divisible by 8? (a) 7110 (b) 7114 (c) 7104 (d) 7099
10. 10.Which of these numbers is divisible by 6? (a) 3393 (b) 3404 (c) 3396 (d) 3401
11. 11.What is the units digit of 8^23? (a) 4 (b) 6 (c) 2 (d) 8
12. 12.Which of these numbers is divisible by 3? (a) 1782 (b) 1781 (c) 1786 (d) 1784
13. 13.What is the remainder when 376 is divided by 5? (a) 4 (b) 1 (c) 0 (d) 2
14. 14.What is the smallest number you must add to 318 to make it divisible by 5? (a) 2 (b) 3 (c) 4 (d) 1
15. 15.Which of these numbers is divisible by 3? (a) 515 (b) 512 (c) 518 (d) 513
16. 16.What is the smallest number you must add to 758 to make it divisible by 12? (a) 10 (b) 2 (c) 11 (d) 9
17. 17.What is the units digit of 8^36? (a) 2 (b) 6 (c) 4 (d) 8
18. 18.What is the remainder when 567 is divided by 11? (a) 8 (b) 7 (c) 6 (d) 5
19. 19.What is the units digit of 9^13? (a) 9 (b) 8 (c) 1 (d) 7
20. 20.What is the smallest number you must add to 231 to make it divisible by 12? (a) 10 (b) 3 (c) 9 (d) 8
21. 21.What is the remainder when 469 is divided by 10? (a) 8 (b) 1 (c) 9 (d) 7
22. 22.What is the remainder when 306 is divided by 5? (a) 1 (b) 0 (c) 4 (d) 2
23. 23.What is the remainder when 344 is divided by 9? (a) 7 (b) 2 (c) 3 (d) 1
24. 24.What is the smallest number you must add to 399 to make it divisible by 11? (a) 8 (b) 3 (c) 7 (d) 9
25. 25.What is the units digit of 2^37? (a) 8 (b) 6 (c) 4 (d) 2
26. 26.What is the smallest number you must add to 396 to make it divisible by 5? (a) 3 (b) 1 (c) 2 (d) 4
27. 27.What is the remainder when 292 is divided by 5? (a) 4 (b) 1 (c) 2 (d) 3
28. 28.Which of these numbers is divisible by 8? (a) 4960 (b) 4966 (c) 4972 (d) 4958
29. 29.What is the remainder when 233 is divided by 4? (a) 1 (b) 3 (c) 0 (d) 2
30. 30.What is the units digit of 7^52? (a) 7 (b) 3 (c) 9 (d) 1
31. 31.What is the remainder when 248 is divided by 12? (a) 7 (b) 8 (c) 9 (d) 4
32. 32.What is the remainder when 150 is divided by 7? (a) 4 (b) 3 (c) 5 (d) 2
33. 33.What is the units digit of 9^50? (a) 2 (b) 9 (c) 1 (d) 0
34. 34.Which of these numbers is divisible by 6? (a) 2249 (b) 2251 (c) 2261 (d) 2250
35. 35.What is the remainder when 182 is divided by 3? (a) 2 (b) 1 (c) 0
36. 36.What is the units digit of 7^46? (a) 1 (b) 9 (c) 7 (d) 3
37. 37.What is the units digit of 4^59? (a) 3 (b) 4 (c) 6 (d) 5
38. 38.What is the remainder when 795 is divided by 10? (a) 7 (b) 5 (c) 6 (d) 4
39. 39.What is the remainder when 179 is divided by 6? (a) 4 (b) 1 (c) 3 (d) 5
40. 40.What is the smallest number you must add to 311 to make it divisible by 10? (a) 8 (b) 9 (c) 7 (d) 1

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