Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 636 · 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 smallest number you must add to 529 to make it divisible by 9? (a) 3 (b) 7 (c) 2 (d) 1
2. 2.What is the smallest number you must add to 771 to make it divisible by 11? (a) 8 (b) 10 (c) 1 (d) 9
3. 3.What is the units digit of 9^49? (a) 7 (b) 1 (c) 9 (d) 8
4. 4.Which of these numbers is divisible by 6? (a) 5127 (b) 5130 (c) 5134 (d) 5137
5. 5.Which of these numbers is divisible by 8? (a) 2040 (b) 2042 (c) 2053 (d) 2037
6. 6.What is the remainder when 239 is divided by 3? (a) 0 (b) 2 (c) 1
7. 7.What is the remainder when 86 is divided by 7? (a) 1 (b) 3 (c) 5 (d) 2
8. 8.What is the units digit of 7^22? (a) 1 (b) 9 (c) 3 (d) 7
9. 9.Which of these numbers is divisible by 6? (a) 3962 (b) 3960 (c) 3958 (d) 3968
10. 10.What is the units digit of 4^59? (a) 3 (b) 5 (c) 4 (d) 6
11. 11.What is the remainder when 217 is divided by 5? (a) 3 (b) 4 (c) 1 (d) 2
12. 12.What is the units digit of 2^20? (a) 4 (b) 6 (c) 2 (d) 8
13. 13.What is the remainder when 592 is divided by 12? (a) 3 (b) 5 (c) 8 (d) 4
14. 14.What is the units digit of 2^28? (a) 8 (b) 6 (c) 2 (d) 4
15. 15.What is the remainder when 167 is divided by 4? (a) 1 (b) 0 (c) 2 (d) 3
16. 16.What is the remainder when 406 is divided by 8? (a) 7 (b) 2 (c) 5 (d) 6
17. 17.What is the smallest number you must add to 285 to make it divisible by 4? (a) 1 (b) 2 (c) 3
18. 18.What is the units digit of 2^11? (a) 8 (b) 4 (c) 6 (d) 2
19. 19.What is the units digit of 8^49? (a) 2 (b) 6 (c) 8 (d) 4
20. 20.Which of these numbers is divisible by 3? (a) 2596 (b) 2594 (c) 2595 (d) 2599
21. 21.What is the units digit of 8^15? (a) 8 (b) 6 (c) 2 (d) 4
22. 22.Which of these numbers is divisible by 3? (a) 2371 (b) 2375 (c) 2377 (d) 2373
23. 23.What is the smallest number you must add to 868 to make it divisible by 11? (a) 1 (b) 3 (c) 10 (d) 2
24. 24.What is the remainder when 212 is divided by 11? (a) 3 (b) 4 (c) 2 (d) 8
25. 25.What is the units digit of 8^12? (a) 8 (b) 2 (c) 4 (d) 6
26. 26.What is the remainder when 194 is divided by 5? (a) 3 (b) 2 (c) 1 (d) 4
27. 27.What is the units digit of 8^29? (a) 6 (b) 4 (c) 8 (d) 2
28. 28.Which of these numbers is divisible by 4? (a) 922 (b) 918 (c) 920 (d) 927
29. 29.What is the remainder when 101 is divided by 4? (a) 3 (b) 2 (c) 0 (d) 1
30. 30.What is the remainder when 505 is divided by 11? (a) 9 (b) 1 (c) 8 (d) 10
31. 31.Which of these numbers is divisible by 4? (a) 613 (b) 616 (c) 623 (d) 617
32. 32.What is the smallest number you must add to 521 to make it divisible by 11? (a) 7 (b) 8 (c) 4 (d) 6
33. 33.What is the units digit of 8^14? (a) 8 (b) 2 (c) 6 (d) 4
34. 34.What is the remainder when 87 is divided by 5? (a) 3 (b) 1 (c) 4 (d) 2
35. 35.What is the remainder when 89 is divided by 6? (a) 1 (b) 3 (c) 5 (d) 4
36. 36.Which of these numbers is divisible by 4? (a) 648 (b) 651 (c) 655 (d) 647
37. 37.Which of these numbers is divisible by 8? (a) 4221 (b) 4238 (c) 4224 (d) 4226
38. 38.What is the remainder when 262 is divided by 8? (a) 7 (b) 2 (c) 5 (d) 6
39. 39.What is the units digit of 8^31? (a) 8 (b) 4 (c) 6 (d) 2
40. 40.What is the remainder when 134 is divided by 11? (a) 9 (b) 2 (c) 3 (d) 1

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