Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 568 · 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 562 is divided by 7? (a) 5 (b) 2 (c) 1 (d) 3
2. 2.What is the units digit of 8^16? (a) 2 (b) 6 (c) 8 (d) 4
3. 3.What is the units digit of 3^45? (a) 1 (b) 3 (c) 9 (d) 7
4. 4.Which of these numbers is divisible by 8? (a) 5037 (b) 5041 (c) 5032 (d) 5029
5. 5.What is the remainder when 282 is divided by 4? (a) 2 (b) 1 (c) 3 (d) 0
6. 6.What is the remainder when 149 is divided by 6? (a) 3 (b) 4 (c) 1 (d) 5
7. 7.Which of these numbers is divisible by 11? (a) 4022 (b) 4039 (c) 4026 (d) 4033
8. 8.What is the smallest number you must add to 435 to make it divisible by 10? (a) 4 (b) 7 (c) 5 (d) 6
9. 9.Which of these numbers is divisible by 8? (a) 1848 (b) 1846 (c) 1855 (d) 1861
10. 10.What is the smallest number you must add to 143 to make it divisible by 8? (a) 7 (b) 1 (c) 3 (d) 2
11. 11.What is the remainder when 88 is divided by 3? (a) 2 (b) 1 (c) 0
12. 12.What is the remainder when 507 is divided by 10? (a) 6 (b) 7 (c) 8 (d) 3
13. 13.What is the smallest number you must add to 760 to make it divisible by 11? (a) 8 (b) 10 (c) 9 (d) 1
14. 14.What is the units digit of 7^37? (a) 3 (b) 7 (c) 1 (d) 9
15. 15.What is the smallest number you must add to 281 to make it divisible by 9? (a) 6 (b) 8 (c) 7 (d) 2
16. 16.Which of these numbers is divisible by 6? (a) 4095 (b) 4086 (c) 4081 (d) 4087
17. 17.What is the smallest number you must add to 766 to make it divisible by 12? (a) 3 (b) 2 (c) 10 (d) 1
18. 18.Which of these numbers is divisible by 8? (a) 2192 (b) 2205 (c) 2199 (d) 2185
19. 19.What is the units digit of 4^33? (a) 3 (b) 5 (c) 4 (d) 6
20. 20.Which of these numbers is divisible by 8? (a) 6865 (b) 6872 (c) 6874 (d) 6887
21. 21.What is the smallest number you must add to 415 to make it divisible by 11? (a) 3 (b) 4 (c) 8 (d) 2
22. 22.Which of these numbers is divisible by 8? (a) 5215 (b) 5200 (c) 5203 (d) 5193
23. 23.What is the units digit of 3^21? (a) 3 (b) 9 (c) 7 (d) 1
24. 24.What is the remainder when 85 is divided by 6? (a) 1 (b) 5 (c) 0 (d) 2
25. 25.What is the smallest number you must add to 689 to make it divisible by 9? (a) 3 (b) 4 (c) 6 (d) 5
26. 26.What is the remainder when 350 is divided by 12? (a) 2 (b) 3 (c) 10 (d) 1
27. 27.What is the remainder when 227 is divided by 12? (a) 1 (b) 11 (c) 9 (d) 10
28. 28.Which of these numbers is divisible by 9? (a) 4217 (b) 4205 (c) 4203 (d) 4201
29. 29.What is the remainder when 190 is divided by 3? (a) 2 (b) 0 (c) 1
30. 30.What is the remainder when 676 is divided by 10? (a) 7 (b) 4 (c) 6 (d) 5
31. 31.What is the units digit of 4^46? (a) 6 (b) 5 (c) 7 (d) 4
32. 32.Which of these numbers is divisible by 11? (a) 2805 (b) 2823 (c) 2814 (d) 2803
33. 33.What is the remainder when 362 is divided by 5? (a) 1 (b) 4 (c) 2 (d) 3
34. 34.Which of these numbers is divisible by 3? (a) 1912 (b) 1910 (c) 1916 (d) 1911
35. 35.What is the remainder when 101 is divided by 3? (a) 1 (b) 0 (c) 2
36. 36.Which of these numbers is divisible by 6? (a) 1429 (b) 1423 (c) 1422 (d) 1420
37. 37.Which of these numbers is divisible by 9? (a) 2448 (b) 2449 (c) 2461 (d) 2444
38. 38.What is the smallest number you must add to 77 to make it divisible by 5? (a) 1 (b) 4 (c) 2 (d) 3
39. 39.Which of these numbers is divisible by 11? (a) 5629 (b) 5612 (c) 5601 (d) 5610
40. 40.What is the remainder when 349 is divided by 6? (a) 5 (b) 0 (c) 2 (d) 1

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