Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 661 · 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 4? (a) 2041 (b) 2047 (c) 2038 (d) 2040
2. 2.What is the remainder when 243 is divided by 8? (a) 2 (b) 4 (c) 5 (d) 3
3. 3.What is the remainder when 554 is divided by 10? (a) 6 (b) 5 (c) 3 (d) 4
4. 4.What is the remainder when 55 is divided by 4? (a) 0 (b) 1 (c) 3 (d) 2
5. 5.Which of these numbers is divisible by 4? (a) 2783 (b) 2787 (c) 2784 (d) 2789
6. 6.What is the remainder when 736 is divided by 12? (a) 3 (b) 5 (c) 8 (d) 4
7. 7.What is the smallest number you must add to 652 to make it divisible by 11? (a) 7 (b) 9 (c) 3 (d) 8
8. 8.What is the smallest number you must add to 839 to make it divisible by 12? (a) 11 (b) 2 (c) 1 (d) 3
9. 9.What is the smallest number you must add to 742 to make it divisible by 12? (a) 1 (b) 3 (c) 10 (d) 2
10. 10.What is the smallest number you must add to 277 to make it divisible by 5? (a) 1 (b) 2 (c) 3 (d) 4
11. 11.What is the units digit of 9^19? (a) 9 (b) 1 (c) 8 (d) 7
12. 12.Which of these numbers is divisible by 11? (a) 6065 (b) 6050 (c) 6056 (d) 6042
13. 13.What is the remainder when 267 is divided by 6? (a) 4 (b) 2 (c) 5 (d) 3
14. 14.What is the remainder when 259 is divided by 10? (a) 1 (b) 8 (c) 9 (d) 7
15. 15.What is the remainder when 578 is divided by 9? (a) 1 (b) 3 (c) 2 (d) 7
16. 16.What is the smallest number you must add to 363 to make it divisible by 6? (a) 5 (b) 4 (c) 3 (d) 2
17. 17.What is the remainder when 152 is divided by 11? (a) 10 (b) 8 (c) 2 (d) 9
18. 18.What is the units digit of 3^17? (a) 9 (b) 3 (c) 7 (d) 1
19. 19.What is the units digit of 4^21? (a) 5 (b) 3 (c) 6 (d) 4
20. 20.Which of these numbers is divisible by 9? (a) 7828 (b) 7812 (c) 7807 (d) 7816
21. 21.What is the remainder when 722 is divided by 11? (a) 4 (b) 6 (c) 7 (d) 8
22. 22.Which of these numbers is divisible by 6? (a) 2341 (b) 2336 (c) 2349 (d) 2340
23. 23.Which of these numbers is divisible by 6? (a) 695 (b) 680 (c) 684 (d) 687
24. 24.What is the remainder when 292 is divided by 9? (a) 6 (b) 5 (c) 3 (d) 4
25. 25.What is the remainder when 97 is divided by 4? (a) 0 (b) 1 (c) 3 (d) 2
26. 26.What is the remainder when 400 is divided by 7? (a) 0 (b) 6 (c) 1 (d) 2
27. 27.What is the remainder when 122 is divided by 3? (a) 1 (b) 0 (c) 2
28. 28.What is the smallest number you must add to 178 to make it divisible by 3? (a) 2 (b) 1
29. 29.What is the units digit of 3^51? (a) 9 (b) 1 (c) 7 (d) 3
30. 30.What is the remainder when 263 is divided by 9? (a) 7 (b) 2 (c) 1 (d) 3
31. 31.What is the remainder when 371 is divided by 12? (a) 11 (b) 9 (c) 10 (d) 1
32. 32.Which of these numbers is divisible by 8? (a) 647 (b) 655 (c) 636 (d) 640
33. 33.Which of these numbers is divisible by 6? (a) 4045 (b) 4039 (c) 4038 (d) 4036
34. 34.What is the units digit of 9^42? (a) 0 (b) 1 (c) 9 (d) 2
35. 35.What is the units digit of 8^57? (a) 6 (b) 8 (c) 2 (d) 4
36. 36.What is the units digit of 2^45? (a) 6 (b) 4 (c) 8 (d) 2
37. 37.What is the remainder when 107 is divided by 3? (a) 0 (b) 2 (c) 1
38. 38.What is the remainder when 163 is divided by 6? (a) 1 (b) 5 (c) 0 (d) 2
39. 39.Which of these numbers is divisible by 3? (a) 559 (b) 563 (c) 566 (d) 561
40. 40.Which of these numbers is divisible by 11? (a) 3958 (b) 3949 (c) 3945 (d) 3961

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