Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 77 · 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 units digit of 2^39? (a) 6 (b) 4 (c) 8 (d) 2
2. 2.What is the units digit of 7^41? (a) 1 (b) 3 (c) 9 (d) 7
3. 3.What is the smallest number you must add to 202 to make it divisible by 3? (a) 1 (b) 2
4. 4.What is the units digit of 9^60? (a) 0 (b) 1 (c) 9 (d) 2
5. 5.What is the smallest number you must add to 323 to make it divisible by 6? (a) 3 (b) 2 (c) 5 (d) 1
6. 6.Which of these numbers is divisible by 6? (a) 3646 (b) 3636 (c) 3634 (d) 3639
7. 7.What is the units digit of 8^12? (a) 4 (b) 6 (c) 2 (d) 8
8. 8.What is the remainder when 246 is divided by 7? (a) 6 (b) 0 (c) 1 (d) 2
9. 9.What is the units digit of 9^47? (a) 8 (b) 1 (c) 9 (d) 7
10. 10.What is the units digit of 7^52? (a) 9 (b) 7 (c) 1 (d) 3
11. 11.What is the units digit of 7^49? (a) 1 (b) 7 (c) 3 (d) 9
12. 12.Which of these numbers is divisible by 8? (a) 2670 (b) 2660 (c) 2664 (d) 2675
13. 13.What is the smallest number you must add to 519 to make it divisible by 7? (a) 4 (b) 6 (c) 1 (d) 5
14. 14.What is the remainder when 279 is divided by 10? (a) 8 (b) 1 (c) 9 (d) 7
15. 15.What is the units digit of 3^24? (a) 3 (b) 1 (c) 7 (d) 9
16. 16.What is the units digit of 4^50? (a) 5 (b) 6 (c) 4 (d) 7
17. 17.What is the remainder when 554 is divided by 11? (a) 5 (b) 4 (c) 7 (d) 3
18. 18.What is the remainder when 480 is divided by 9? (a) 6 (b) 3 (c) 4 (d) 2
19. 19.What is the remainder when 449 is divided by 11? (a) 10 (b) 8 (c) 9 (d) 2
20. 20.What is the smallest number you must add to 167 to make it divisible by 7? (a) 3 (b) 2 (c) 1 (d) 6
21. 21.What is the units digit of 3^51? (a) 3 (b) 7 (c) 9 (d) 1
22. 22.Which of these numbers is divisible by 6? (a) 5222 (b) 5215 (c) 5231 (d) 5220
23. 23.Which of these numbers is divisible by 9? (a) 675 (b) 680 (c) 690 (d) 669
24. 24.What is the units digit of 7^59? (a) 1 (b) 9 (c) 7 (d) 3
25. 25.What is the remainder when 94 is divided by 4? (a) 1 (b) 2 (c) 0 (d) 3
26. 26.What is the remainder when 278 is divided by 10? (a) 8 (b) 9 (c) 7 (d) 2
27. 27.What is the units digit of 2^17? (a) 4 (b) 2 (c) 6 (d) 8
28. 28.What is the remainder when 232 is divided by 11? (a) 10 (b) 1 (c) 0 (d) 2
29. 29.Which of these numbers is divisible by 3? (a) 2135 (b) 2141 (c) 2136 (d) 2138
30. 30.What is the smallest number you must add to 687 to make it divisible by 9? (a) 3 (b) 5 (c) 7 (d) 6
31. 31.What is the smallest number you must add to 93 to make it divisible by 5? (a) 1 (b) 3 (c) 2 (d) 4
32. 32.What is the units digit of 2^44? (a) 2 (b) 4 (c) 8 (d) 6
33. 33.Which of these numbers is divisible by 9? (a) 6462 (b) 6456 (c) 6470 (d) 6475
34. 34.Which of these numbers is divisible by 6? (a) 803 (b) 796 (c) 798 (d) 808
35. 35.Which of these numbers is divisible by 9? (a) 2808 (b) 2821 (c) 2809 (d) 2805
36. 36.What is the remainder when 230 is divided by 4? (a) 3 (b) 1 (c) 0 (d) 2
37. 37.Which of these numbers is divisible by 11? (a) 2794 (b) 2787 (c) 2801 (d) 2806
38. 38.What is the units digit of 2^51? (a) 8 (b) 6 (c) 4 (d) 2
39. 39.What is the remainder when 133 is divided by 3? (a) 2 (b) 0 (c) 1
40. 40.What is the smallest number you must add to 282 to make it divisible by 5? (a) 4 (b) 1 (c) 2 (d) 3

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