Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 796 · 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 remainder when 184 is divided by 10? (a) 3 (b) 5 (c) 4 (d) 6
2. 2.What is the remainder when 190 is divided by 12? (a) 10 (b) 9 (c) 11 (d) 2
3. 3.Which of these numbers is divisible by 8? (a) 477 (b) 492 (c) 480 (d) 486
4. 4.What is the units digit of 9^29? (a) 1 (b) 9 (c) 8 (d) 7
5. 5.What is the remainder when 323 is divided by 7? (a) 0 (b) 6 (c) 1 (d) 2
6. 6.What is the remainder when 634 is divided by 8? (a) 1 (b) 2 (c) 3 (d) 6
7. 7.What is the smallest number you must add to 194 to make it divisible by 4? (a) 3 (b) 2 (c) 1
8. 8.What is the remainder when 664 is divided by 9? (a) 6 (b) 2 (c) 8 (d) 7
9. 9.What is the smallest number you must add to 280 to make it divisible by 9? (a) 7 (b) 6 (c) 8 (d) 1
10. 10.What is the remainder when 172 is divided by 11? (a) 8 (b) 6 (c) 7 (d) 4
11. 11.Which of these numbers is divisible by 6? (a) 4709 (b) 4698 (c) 4693 (d) 4701
12. 12.What is the remainder when 481 is divided by 9? (a) 5 (b) 4 (c) 6 (d) 3
13. 13.What is the units digit of 9^43? (a) 8 (b) 1 (c) 9 (d) 7
14. 14.Which of these numbers is divisible by 8? (a) 561 (b) 552 (c) 547 (d) 553
15. 15.Which of these numbers is divisible by 3? (a) 1421 (b) 1417 (c) 1419 (d) 1423
16. 16.What is the remainder when 515 is divided by 12? (a) 9 (b) 1 (c) 10 (d) 11
17. 17.Which of these numbers is divisible by 11? (a) 4634 (b) 4624 (c) 4645 (d) 4631
18. 18.Which of these numbers is divisible by 9? (a) 3144 (b) 3135 (c) 3154 (d) 3141
19. 19.Which of these numbers is divisible by 11? (a) 7326 (b) 7339 (c) 7316 (d) 7333
20. 20.Which of these numbers is divisible by 6? (a) 2628 (b) 2633 (c) 2639 (d) 2627
21. 21.What is the units digit of 4^38? (a) 6 (b) 7 (c) 5 (d) 4
22. 22.What is the units digit of 8^45? (a) 6 (b) 8 (c) 2 (d) 4
23. 23.Which of these numbers is divisible by 6? (a) 3872 (b) 3867 (c) 3879 (d) 3870
24. 24.What is the remainder when 352 is divided by 5? (a) 4 (b) 1 (c) 2 (d) 3
25. 25.Which of these numbers is divisible by 6? (a) 4336 (b) 4331 (c) 4342 (d) 4332
26. 26.What is the remainder when 100 is divided by 3? (a) 2 (b) 0 (c) 1
27. 27.Which of these numbers is divisible by 6? (a) 5082 (b) 5079 (c) 5087 (d) 5089
28. 28.Which of these numbers is divisible by 8? (a) 1640 (b) 1649 (c) 1637 (d) 1642
29. 29.What is the units digit of 9^38? (a) 0 (b) 2 (c) 1 (d) 9
30. 30.What is the smallest number you must add to 251 to make it divisible by 4? (a) 1 (b) 3 (c) 2
31. 31.Which of these numbers is divisible by 6? (a) 4964 (b) 4952 (c) 4959 (d) 4956
32. 32.What is the smallest number you must add to 541 to make it divisible by 8? (a) 3 (b) 5 (c) 2 (d) 4
33. 33.What is the units digit of 3^51? (a) 1 (b) 9 (c) 7 (d) 3
34. 34.Which of these numbers is divisible by 9? (a) 5139 (b) 5133 (c) 5152 (d) 5145
35. 35.What is the smallest number you must add to 795 to make it divisible by 10? (a) 7 (b) 6 (c) 5 (d) 4
36. 36.What is the smallest number you must add to 625 to make it divisible by 12? (a) 10 (b) 1 (c) 9 (d) 11
37. 37.What is the units digit of 3^38? (a) 9 (b) 7 (c) 1 (d) 3
38. 38.What is the remainder when 510 is divided by 7? (a) 4 (b) 6 (c) 1 (d) 5
39. 39.What is the smallest number you must add to 43 to make it divisible by 4? (a) 2 (b) 3 (c) 1
40. 40.What is the units digit of 9^10? (a) 2 (b) 0 (c) 1 (d) 9

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