Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 16 · 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) 847 (b) 850 (c) 844 (d) 842
2. 2.What is the remainder when 471 is divided by 10? (a) 0 (b) 9 (c) 2 (d) 1
3. 3.Which of these numbers is divisible by 4? (a) 287 (b) 293 (c) 290 (d) 288
4. 4.What is the remainder when 594 is divided by 8? (a) 2 (b) 6 (c) 1 (d) 3
5. 5.What is the smallest number you must add to 566 to make it divisible by 11? (a) 8 (b) 5 (c) 6 (d) 7
6. 6.What is the remainder when 151 is divided by 7? (a) 3 (b) 5 (c) 4 (d) 6
7. 7.What is the remainder when 68 is divided by 5? (a) 3 (b) 4 (c) 2 (d) 1
8. 8.What is the smallest number you must add to 300 to make it divisible by 8? (a) 6 (b) 5 (c) 4 (d) 3
9. 9.What is the units digit of 9^39? (a) 7 (b) 1 (c) 9 (d) 8
10. 10.Which of these numbers is divisible by 11? (a) 9730 (b) 9713 (c) 9707 (d) 9715
11. 11.What is the units digit of 4^39? (a) 5 (b) 4 (c) 6 (d) 3
12. 12.What is the units digit of 9^54? (a) 9 (b) 1 (c) 2 (d) 0
13. 13.What is the units digit of 3^26? (a) 3 (b) 7 (c) 1 (d) 9
14. 14.What is the units digit of 7^40? (a) 7 (b) 9 (c) 1 (d) 3
15. 15.What is the smallest number you must add to 243 to make it divisible by 8? (a) 6 (b) 3 (c) 4 (d) 5
16. 16.Which of these numbers is divisible by 6? (a) 4551 (b) 4557 (c) 4561 (d) 4554
17. 17.Which of these numbers is divisible by 11? (a) 8798 (b) 8789 (c) 8805 (d) 8787
18. 18.What is the units digit of 2^32? (a) 8 (b) 6 (c) 2 (d) 4
19. 19.What is the remainder when 161 is divided by 4? (a) 2 (b) 3 (c) 0 (d) 1
20. 20.What is the remainder when 62 is divided by 3? (a) 0 (b) 1 (c) 2
21. 21.Which of these numbers is divisible by 4? (a) 551 (b) 548 (c) 547 (d) 553
22. 22.Which of these numbers is divisible by 6? (a) 1720 (b) 1725 (c) 1722 (d) 1732
23. 23.What is the units digit of 8^36? (a) 2 (b) 6 (c) 8 (d) 4
24. 24.Which of these numbers is divisible by 3? (a) 2444 (b) 2441 (c) 2439 (d) 2438
25. 25.What is the remainder when 589 is divided by 11? (a) 6 (b) 7 (c) 8 (d) 5
26. 26.What is the remainder when 356 is divided by 11? (a) 5 (b) 7 (c) 3 (d) 4
27. 27.Which of these numbers is divisible by 3? (a) 661 (b) 665 (c) 658 (d) 660
28. 28.What is the units digit of 2^57? (a) 8 (b) 6 (c) 2 (d) 4
29. 29.What is the units digit of 2^28? (a) 6 (b) 4 (c) 2 (d) 8
30. 30.Which of these numbers is divisible by 8? (a) 1776 (b) 1772 (c) 1786 (d) 1778
31. 31.What is the units digit of 3^51? (a) 1 (b) 9 (c) 3 (d) 7
32. 32.Which of these numbers is divisible by 3? (a) 2072 (b) 2078 (c) 2073 (d) 2074
33. 33.Which of these numbers is divisible by 9? (a) 7899 (b) 7884 (c) 7888 (d) 7879
34. 34.What is the remainder when 609 is divided by 10? (a) 9 (b) 8 (c) 1 (d) 7
35. 35.What is the smallest number you must add to 570 to make it divisible by 9? (a) 3 (b) 5 (c) 6 (d) 7
36. 36.What is the smallest number you must add to 261 to make it divisible by 11? (a) 4 (b) 2 (c) 3 (d) 8
37. 37.Which of these numbers is divisible by 11? (a) 2992 (b) 3008 (c) 3000 (d) 2986
38. 38.What is the remainder when 146 is divided by 4? (a) 3 (b) 0 (c) 1 (d) 2
39. 39.Which of these numbers is divisible by 3? (a) 2362 (b) 2368 (c) 2366 (d) 2364
40. 40.What is the smallest number you must add to 212 to make it divisible by 10? (a) 2 (b) 7 (c) 9 (d) 8

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