Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 49 · 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) 2100 (b) 2101 (c) 2099 (d) 2106
2. 2.What is the remainder when 197 is divided by 5? (a) 4 (b) 1 (c) 2 (d) 3
3. 3.Which of these numbers is divisible by 8? (a) 4635 (b) 4624 (c) 4617 (d) 4625
4. 4.What is the smallest number you must add to 201 to make it divisible by 11? (a) 9 (b) 8 (c) 7 (d) 3
5. 5.Which of these numbers is divisible by 11? (a) 4855 (b) 4877 (c) 4862 (d) 4865
6. 6.What is the smallest number you must add to 74 to make it divisible by 4? (a) 1 (b) 3 (c) 2
7. 7.Which of these numbers is divisible by 3? (a) 2609 (b) 2605 (c) 2607 (d) 2611
8. 8.What is the smallest number you must add to 553 to make it divisible by 9? (a) 6 (b) 7 (c) 4 (d) 5
9. 9.What is the remainder when 136 is divided by 12? (a) 3 (b) 5 (c) 4 (d) 8
10. 10.Which of these numbers is divisible by 11? (a) 7699 (b) 7689 (c) 7684 (d) 7703
11. 11.What is the units digit of 9^57? (a) 8 (b) 7 (c) 1 (d) 9
12. 12.What is the units digit of 8^46? (a) 4 (b) 2 (c) 8 (d) 6
13. 13.What is the remainder when 390 is divided by 11? (a) 4 (b) 5 (c) 6 (d) 7
14. 14.What is the units digit of 2^11? (a) 4 (b) 6 (c) 8 (d) 2
15. 15.What is the remainder when 405 is divided by 10? (a) 6 (b) 5 (c) 4 (d) 7
16. 16.Which of these numbers is divisible by 11? (a) 1042 (b) 1062 (c) 1045 (d) 1051
17. 17.Which of these numbers is divisible by 11? (a) 1475 (b) 1485 (c) 1490 (d) 1497
18. 18.Which of these numbers is divisible by 4? (a) 3441 (b) 3445 (c) 3449 (d) 3444
19. 19.Which of these numbers is divisible by 8? (a) 4984 (b) 4991 (c) 4999 (d) 4981
20. 20.Which of these numbers is divisible by 9? (a) 6369 (b) 6363 (c) 6380 (d) 6357
21. 21.What is the smallest number you must add to 299 to make it divisible by 4? (a) 3 (b) 2 (c) 1
22. 22.What is the smallest number you must add to 118 to make it divisible by 6? (a) 3 (b) 1 (c) 4 (d) 2
23. 23.Which of these numbers is divisible by 4? (a) 1879 (b) 1874 (c) 1872 (d) 1870
24. 24.Which of these numbers is divisible by 3? (a) 683 (b) 680 (c) 678 (d) 677
25. 25.Which of these numbers is divisible by 3? (a) 1204 (b) 1206 (c) 1208 (d) 1211
26. 26.What is the remainder when 105 is divided by 8? (a) 2 (b) 7 (c) 1 (d) 0
27. 27.What is the smallest number you must add to 138 to make it divisible by 5? (a) 1 (b) 3 (c) 4 (d) 2
28. 28.What is the smallest number you must add to 249 to make it divisible by 6? (a) 5 (b) 4 (c) 3 (d) 2
29. 29.Which of these numbers is divisible by 9? (a) 3971 (b) 3986 (c) 3993 (d) 3978
30. 30.What is the remainder when 143 is divided by 3? (a) 1 (b) 0 (c) 2
31. 31.What is the smallest number you must add to 556 to make it divisible by 12? (a) 4 (b) 7 (c) 9 (d) 8
32. 32.What is the units digit of 4^22? (a) 4 (b) 7 (c) 6 (d) 5
33. 33.What is the units digit of 7^14? (a) 7 (b) 3 (c) 1 (d) 9
34. 34.Which of these numbers is divisible by 8? (a) 4804 (b) 4808 (c) 4809 (d) 4818
35. 35.What is the remainder when 287 is divided by 5? (a) 1 (b) 2 (c) 3 (d) 4
36. 36.What is the remainder when 905 is divided by 12? (a) 5 (b) 4 (c) 6 (d) 7
37. 37.Which of these numbers is divisible by 8? (a) 5690 (b) 5696 (c) 5702 (d) 5707
38. 38.What is the remainder when 475 is divided by 7? (a) 4 (b) 5 (c) 1 (d) 6
39. 39.What is the units digit of 8^41? (a) 4 (b) 8 (c) 6 (d) 2
40. 40.What is the remainder when 276 is divided by 9? (a) 7 (b) 3 (c) 5 (d) 6

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