Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 676 · 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 units digit of 3^41? (a) 9 (b) 3 (c) 1 (d) 7
2. 2.What is the units digit of 8^25? (a) 6 (b) 2 (c) 4 (d) 8
3. 3.Which of these numbers is divisible by 11? (a) 3913 (b) 3896 (c) 3905 (d) 3918
4. 4.What is the units digit of 2^56? (a) 8 (b) 2 (c) 6 (d) 4
5. 5.What is the units digit of 2^29? (a) 2 (b) 6 (c) 8 (d) 4
6. 6.Which of these numbers is divisible by 6? (a) 3368 (b) 3361 (c) 3366 (d) 3374
7. 7.What is the smallest number you must add to 118 to make it divisible by 6? (a) 3 (b) 2 (c) 1 (d) 4
8. 8.What is the units digit of 8^46? (a) 8 (b) 4 (c) 6 (d) 2
9. 9.What is the remainder when 193 is divided by 7? (a) 5 (b) 4 (c) 6 (d) 3
10. 10.What is the remainder when 207 is divided by 6? (a) 5 (b) 2 (c) 3 (d) 4
11. 11.What is the units digit of 2^18? (a) 4 (b) 6 (c) 2 (d) 8
12. 12.What is the remainder when 130 is divided by 3? (a) 2 (b) 0 (c) 1
13. 13.What is the remainder when 529 is divided by 12? (a) 0 (b) 11 (c) 1 (d) 2
14. 14.What is the smallest number you must add to 103 to make it divisible by 8? (a) 7 (b) 3 (c) 2 (d) 1
15. 15.Which of these numbers is divisible by 3? (a) 1879 (b) 1876 (c) 1883 (d) 1878
16. 16.What is the smallest number you must add to 867 to make it divisible by 12? (a) 10 (b) 8 (c) 9 (d) 3
17. 17.Which of these numbers is divisible by 11? (a) 4293 (b) 4269 (c) 4284 (d) 4279
18. 18.What is the remainder when 106 is divided by 4? (a) 2 (b) 1 (c) 3 (d) 0
19. 19.What is the units digit of 2^43? (a) 4 (b) 8 (c) 2 (d) 6
20. 20.Which of these numbers is divisible by 9? (a) 893 (b) 886 (c) 904 (d) 891
21. 21.What is the remainder when 83 is divided by 8? (a) 4 (b) 5 (c) 3 (d) 2
22. 22.Which of these numbers is divisible by 8? (a) 6953 (b) 6965 (c) 6960 (d) 6975
23. 23.What is the remainder when 191 is divided by 8? (a) 1 (b) 7 (c) 5 (d) 6
24. 24.What is the smallest number you must add to 205 to make it divisible by 3? (a) 1 (b) 2
25. 25.What is the smallest number you must add to 542 to make it divisible by 9? (a) 6 (b) 8 (c) 2 (d) 7
26. 26.What is the units digit of 7^23? (a) 9 (b) 3 (c) 1 (d) 7
27. 27.What is the smallest number you must add to 117 to make it divisible by 4? (a) 2 (b) 1 (c) 3
28. 28.What is the remainder when 187 is divided by 4? (a) 1 (b) 0 (c) 3 (d) 2
29. 29.What is the remainder when 266 is divided by 8? (a) 1 (b) 3 (c) 6 (d) 2
30. 30.What is the smallest number you must add to 422 to make it divisible by 7? (a) 4 (b) 5 (c) 6 (d) 2
31. 31.Which of these numbers is divisible by 11? (a) 1131 (b) 1134 (c) 1145 (d) 1133
32. 32.Which of these numbers is divisible by 3? (a) 1168 (b) 1163 (c) 1164 (d) 1166
33. 33.Which of these numbers is divisible by 3? (a) 1663 (b) 1661 (c) 1662 (d) 1667
34. 34.Which of these numbers is divisible by 8? (a) 6009 (b) 5998 (c) 6000 (d) 6002
35. 35.What is the smallest number you must add to 458 to make it divisible by 10? (a) 2 (b) 1 (c) 3 (d) 8
36. 36.What is the units digit of 3^29? (a) 9 (b) 3 (c) 1 (d) 7
37. 37.What is the smallest number you must add to 190 to make it divisible by 7? (a) 5 (b) 6 (c) 1 (d) 4
38. 38.Which of these numbers is divisible by 3? (a) 2401 (b) 2407 (c) 2403 (d) 2405
39. 39.Which of these numbers is divisible by 3? (a) 2442 (b) 2440 (c) 2447 (d) 2444
40. 40.What is the units digit of 2^57? (a) 4 (b) 2 (c) 6 (d) 8

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