Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 740 · 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 3? (a) 533 (b) 535 (c) 539 (d) 534
2. 2.What is the remainder when 320 is divided by 6? (a) 3 (b) 2 (c) 4 (d) 1
3. 3.Which of these numbers is divisible by 8? (a) 6326 (b) 6330 (c) 6315 (d) 6320
4. 4.Which of these numbers is divisible by 6? (a) 5015 (b) 5004 (c) 4999 (d) 5009
5. 5.What is the remainder when 242 is divided by 5? (a) 2 (b) 3 (c) 1 (d) 4
6. 6.What is the remainder when 551 is divided by 11? (a) 1 (b) 0 (c) 10 (d) 2
7. 7.What is the units digit of 4^60? (a) 5 (b) 7 (c) 4 (d) 6
8. 8.What is the remainder when 548 is divided by 8? (a) 5 (b) 3 (c) 6 (d) 4
9. 9.Which of these numbers is divisible by 11? (a) 6005 (b) 5995 (c) 6016 (d) 5987
10. 10.What is the units digit of 9^24? (a) 9 (b) 1 (c) 2 (d) 0
11. 11.What is the smallest number you must add to 194 to make it divisible by 8? (a) 7 (b) 5 (c) 2 (d) 6
12. 12.What is the smallest number you must add to 517 to make it divisible by 9? (a) 5 (b) 7 (c) 4 (d) 6
13. 13.What is the remainder when 767 is divided by 11? (a) 7 (b) 8 (c) 9 (d) 3
14. 14.What is the remainder when 630 is divided by 8? (a) 7 (b) 5 (c) 2 (d) 6
15. 15.What is the smallest number you must add to 474 to make it divisible by 7? (a) 5 (b) 2 (c) 3 (d) 1
16. 16.What is the remainder when 355 is divided by 9? (a) 4 (b) 3 (c) 6 (d) 5
17. 17.Which of these numbers is divisible by 4? (a) 1767 (b) 1773 (c) 1768 (d) 1769
18. 18.What is the units digit of 3^21? (a) 1 (b) 3 (c) 9 (d) 7
19. 19.Which of these numbers is divisible by 9? (a) 6240 (b) 6233 (c) 6228 (d) 6224
20. 20.Which of these numbers is divisible by 6? (a) 432 (b) 427 (c) 437 (d) 441
21. 21.What is the remainder when 303 is divided by 4? (a) 1 (b) 3 (c) 2 (d) 0
22. 22.Which of these numbers is divisible by 8? (a) 1924 (b) 1914 (c) 1933 (d) 1920
23. 23.Which of these numbers is divisible by 4? (a) 492 (b) 491 (c) 498 (d) 495
24. 24.Which of these numbers is divisible by 8? (a) 2138 (b) 2136 (c) 2150 (d) 2134
25. 25.What is the remainder when 708 is divided by 9? (a) 5 (b) 6 (c) 7 (d) 3
26. 26.What is the smallest number you must add to 79 to make it divisible by 5? (a) 1 (b) 3 (c) 4 (d) 2
27. 27.Which of these numbers is divisible by 4? (a) 2484 (b) 2485 (c) 2490 (d) 2483
28. 28.What is the units digit of 9^57? (a) 9 (b) 7 (c) 8 (d) 1
29. 29.Which of these numbers is divisible by 8? (a) 1816 (b) 1825 (c) 1820 (d) 1809
30. 30.What is the remainder when 151 is divided by 4? (a) 0 (b) 1 (c) 2 (d) 3
31. 31.What is the smallest number you must add to 641 to make it divisible by 12? (a) 5 (b) 8 (c) 6 (d) 7
32. 32.Which of these numbers is divisible by 9? (a) 2633 (b) 2641 (c) 2637 (d) 2647
33. 33.Which of these numbers is divisible by 9? (a) 4662 (b) 4674 (c) 4658 (d) 4669
34. 34.What is the smallest number you must add to 223 to make it divisible by 12? (a) 5 (b) 7 (c) 4 (d) 6
35. 35.What is the units digit of 2^12? (a) 2 (b) 4 (c) 8 (d) 6
36. 36.What is the remainder when 176 is divided by 3? (a) 2 (b) 1 (c) 0
37. 37.Which of these numbers is divisible by 8? (a) 5068 (b) 5075 (c) 5064 (d) 5060
38. 38.What is the units digit of 4^54? (a) 4 (b) 7 (c) 6 (d) 5
39. 39.What is the remainder when 583 is divided by 10? (a) 4 (b) 2 (c) 7 (d) 3
40. 40.What is the smallest number you must add to 194 to make it divisible by 3? (a) 1 (b) 2

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