Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 722 · 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 844 to make it divisible by 11? (a) 2 (b) 8 (c) 3 (d) 4
2. 2.Which of these numbers is divisible by 4? (a) 492 (b) 489 (c) 498 (d) 494
3. 3.What is the remainder when 150 is divided by 7? (a) 3 (b) 4 (c) 5 (d) 2
4. 4.Which of these numbers is divisible by 4? (a) 2674 (b) 2678 (c) 2676 (d) 2681
5. 5.What is the remainder when 297 is divided by 5? (a) 1 (b) 4 (c) 2 (d) 3
6. 6.What is the units digit of 3^28? (a) 3 (b) 9 (c) 7 (d) 1
7. 7.Which of these numbers is divisible by 9? (a) 5874 (b) 5868 (c) 5878 (d) 5862
8. 8.What is the remainder when 133 is divided by 3? (a) 0 (b) 1 (c) 2
9. 9.What is the units digit of 2^52? (a) 2 (b) 4 (c) 8 (d) 6
10. 10.What is the remainder when 194 is divided by 4? (a) 2 (b) 1 (c) 3 (d) 0
11. 11.What is the units digit of 7^15? (a) 1 (b) 7 (c) 3 (d) 9
12. 12.What is the smallest number you must add to 109 to make it divisible by 3? (a) 1 (b) 2
13. 13.What is the units digit of 7^25? (a) 1 (b) 7 (c) 3 (d) 9
14. 14.What is the smallest number you must add to 235 to make it divisible by 3? (a) 2 (b) 1
15. 15.What is the units digit of 3^25? (a) 1 (b) 3 (c) 9 (d) 7
16. 16.Which of these numbers is divisible by 9? (a) 3766 (b) 3752 (c) 3761 (d) 3753
17. 17.What is the smallest number you must add to 361 to make it divisible by 5? (a) 3 (b) 2 (c) 1 (d) 4
18. 18.What is the smallest number you must add to 689 to make it divisible by 10? (a) 1 (b) 2 (c) 3 (d) 9
19. 19.What is the remainder when 723 is divided by 10? (a) 7 (b) 3 (c) 4 (d) 2
20. 20.Which of these numbers is divisible by 4? (a) 755 (b) 747 (c) 748 (d) 750
21. 21.Which of these numbers is divisible by 11? (a) 5320 (b) 5308 (c) 5334 (d) 5313
22. 22.What is the remainder when 146 is divided by 9? (a) 1 (b) 2 (c) 7 (d) 3
23. 23.Which of these numbers is divisible by 9? (a) 6588 (b) 6598 (c) 6589 (d) 6583
24. 24.What is the smallest number you must add to 121 to make it divisible by 9? (a) 4 (b) 5 (c) 7 (d) 6
25. 25.What is the units digit of 4^18? (a) 7 (b) 4 (c) 5 (d) 6
26. 26.Which of these numbers is divisible by 11? (a) 6256 (b) 6237 (c) 6234 (d) 6242
27. 27.What is the remainder when 217 is divided by 9? (a) 2 (b) 0 (c) 1 (d) 8
28. 28.Which of these numbers is divisible by 6? (a) 2609 (b) 2611 (c) 2600 (d) 2604
29. 29.What is the smallest number you must add to 302 to make it divisible by 5? (a) 4 (b) 1 (c) 3 (d) 2
30. 30.What is the smallest number you must add to 265 to make it divisible by 4? (a) 3 (b) 1 (c) 2
31. 31.Which of these numbers is divisible by 11? (a) 5809 (b) 5822 (c) 5808 (d) 5806
32. 32.Which of these numbers is divisible by 11? (a) 1990 (b) 2001 (c) 1972 (d) 1980
33. 33.What is the units digit of 7^54? (a) 7 (b) 3 (c) 1 (d) 9
34. 34.What is the remainder when 351 is divided by 6? (a) 4 (b) 2 (c) 3 (d) 5
35. 35.What is the units digit of 8^15? (a) 6 (b) 4 (c) 2 (d) 8
36. 36.What is the remainder when 322 is divided by 12? (a) 11 (b) 9 (c) 2 (d) 10
37. 37.What is the smallest number you must add to 179 to make it divisible by 3? (a) 2 (b) 1
38. 38.What is the remainder when 806 is divided by 10? (a) 6 (b) 5 (c) 4 (d) 7
39. 39.What is the remainder when 554 is divided by 8? (a) 6 (b) 1 (c) 2 (d) 3
40. 40.What is the remainder when 209 is divided by 7? (a) 5 (b) 1 (c) 4 (d) 6

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