Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 203 · 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 8? (a) 2832 (b) 2838 (c) 2827 (d) 2843
2. 2.Which of these numbers is divisible by 9? (a) 3337 (b) 3351 (c) 3339 (d) 3345
3. 3.Which of these numbers is divisible by 8? (a) 3477 (b) 3466 (c) 3460 (d) 3464
4. 4.What is the remainder when 159 is divided by 8? (a) 1 (b) 5 (c) 6 (d) 7
5. 5.What is the remainder when 223 is divided by 6? (a) 0 (b) 2 (c) 1 (d) 5
6. 6.Which of these numbers is divisible by 6? (a) 1670 (b) 1662 (c) 1659 (d) 1664
7. 7.What is the units digit of 3^35? (a) 1 (b) 9 (c) 3 (d) 7
8. 8.What is the smallest number you must add to 358 to make it divisible by 7? (a) 1 (b) 4 (c) 6 (d) 5
9. 9.Which of these numbers is divisible by 3? (a) 2700 (b) 2699 (c) 2705 (d) 2701
10. 10.What is the units digit of 3^50? (a) 7 (b) 3 (c) 1 (d) 9
11. 11.What is the units digit of 3^28? (a) 3 (b) 9 (c) 1 (d) 7
12. 12.Which of these numbers is divisible by 11? (a) 1533 (b) 1544 (c) 1556 (d) 1540
13. 13.What is the remainder when 147 is divided by 8? (a) 3 (b) 5 (c) 2 (d) 4
14. 14.What is the smallest number you must add to 164 to make it divisible by 3? (a) 2 (b) 1
15. 15.What is the smallest number you must add to 214 to make it divisible by 3? (a) 2 (b) 1
16. 16.What is the remainder when 537 is divided by 7? (a) 2 (b) 6 (c) 5 (d) 4
17. 17.What is the remainder when 133 is divided by 12? (a) 1 (b) 2 (c) 0 (d) 11
18. 18.What is the remainder when 394 is divided by 10? (a) 4 (b) 5 (c) 3 (d) 6
19. 19.Which of these numbers is divisible by 6? (a) 4451 (b) 4454 (c) 4445 (d) 4446
20. 20.What is the smallest number you must add to 325 to make it divisible by 9? (a) 1 (b) 7 (c) 6 (d) 8
21. 21.What is the units digit of 2^45? (a) 2 (b) 8 (c) 6 (d) 4
22. 22.What is the smallest number you must add to 478 to make it divisible by 8? (a) 3 (b) 1 (c) 2 (d) 6
23. 23.What is the remainder when 153 is divided by 5? (a) 4 (b) 2 (c) 1 (d) 3
24. 24.Which of these numbers is divisible by 6? (a) 4040 (b) 4051 (c) 4044 (d) 4045
25. 25.What is the units digit of 8^13? (a) 8 (b) 4 (c) 6 (d) 2
26. 26.What is the units digit of 3^55? (a) 1 (b) 7 (c) 9 (d) 3
27. 27.What is the units digit of 2^34? (a) 2 (b) 6 (c) 8 (d) 4
28. 28.What is the remainder when 593 is divided by 11? (a) 1 (b) 10 (c) 8 (d) 9
29. 29.Which of these numbers is divisible by 8? (a) 1344 (b) 1353 (c) 1351 (d) 1343
30. 30.Which of these numbers is divisible by 11? (a) 1868 (b) 1852 (c) 1848 (d) 1844
31. 31.What is the units digit of 2^55? (a) 4 (b) 2 (c) 6 (d) 8
32. 32.What is the smallest number you must add to 755 to make it divisible by 10? (a) 7 (b) 4 (c) 5 (d) 6
33. 33.Which of these numbers is divisible by 11? (a) 3670 (b) 3663 (c) 3661 (d) 3676
34. 34.What is the smallest number you must add to 83 to make it divisible by 8? (a) 6 (b) 3 (c) 4 (d) 5
35. 35.What is the remainder when 195 is divided by 8? (a) 2 (b) 4 (c) 5 (d) 3
36. 36.What is the smallest number you must add to 259 to make it divisible by 5? (a) 2 (b) 3 (c) 4 (d) 1
37. 37.Which of these numbers is divisible by 3? (a) 401 (b) 404 (c) 397 (d) 399
38. 38.What is the units digit of 2^19? (a) 8 (b) 2 (c) 6 (d) 4
39. 39.What is the units digit of 4^48? (a) 4 (b) 7 (c) 5 (d) 6
40. 40.What is the smallest number you must add to 181 to make it divisible by 4? (a) 1 (b) 2 (c) 3

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