Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 731 · 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 7^12? (a) 7 (b) 1 (c) 9 (d) 3
2. 2.What is the remainder when 271 is divided by 9? (a) 8 (b) 2 (c) 0 (d) 1
3. 3.What is the remainder when 410 is divided by 8? (a) 2 (b) 3 (c) 1 (d) 6
4. 4.What is the smallest number you must add to 236 to make it divisible by 10? (a) 5 (b) 4 (c) 6 (d) 3
5. 5.What is the units digit of 2^46? (a) 6 (b) 2 (c) 4 (d) 8
6. 6.Which of these numbers is divisible by 6? (a) 5005 (b) 4994 (c) 5003 (d) 4998
7. 7.What is the units digit of 3^45? (a) 9 (b) 7 (c) 3 (d) 1
8. 8.Which of these numbers is divisible by 8? (a) 2072 (b) 2082 (c) 2069 (d) 2074
9. 9.Which of these numbers is divisible by 4? (a) 3122 (b) 3118 (c) 3115 (d) 3116
10. 10.Which of these numbers is divisible by 11? (a) 7523 (b) 7527 (c) 7541 (d) 7524
11. 11.What is the units digit of 4^37? (a) 5 (b) 3 (c) 6 (d) 4
12. 12.What is the smallest number you must add to 92 to make it divisible by 9? (a) 2 (b) 8 (c) 6 (d) 7
13. 13.What is the remainder when 113 is divided by 4? (a) 2 (b) 3 (c) 0 (d) 1
14. 14.What is the smallest number you must add to 253 to make it divisible by 8? (a) 2 (b) 3 (c) 5 (d) 4
15. 15.Which of these numbers is divisible by 6? (a) 3661 (b) 3654 (c) 3649 (d) 3658
16. 16.Which of these numbers is divisible by 4? (a) 2349 (b) 2347 (c) 2342 (d) 2344
17. 17.Which of these numbers is divisible by 3? (a) 388 (b) 392 (c) 386 (d) 387
18. 18.What is the smallest number you must add to 428 to make it divisible by 12? (a) 8 (b) 3 (c) 4 (d) 5
19. 19.What is the smallest number you must add to 550 to make it divisible by 8? (a) 6 (b) 1 (c) 2 (d) 3
20. 20.Which of these numbers is divisible by 8? (a) 6515 (b) 6508 (c) 6504 (d) 6501
21. 21.What is the units digit of 9^12? (a) 2 (b) 9 (c) 0 (d) 1
22. 22.What is the remainder when 266 is divided by 11? (a) 1 (b) 2 (c) 9 (d) 3
23. 23.What is the remainder when 365 is divided by 12? (a) 6 (b) 5 (c) 4 (d) 7
24. 24.What is the smallest number you must add to 187 to make it divisible by 10? (a) 7 (b) 2 (c) 4 (d) 3
25. 25.Which of these numbers is divisible by 4? (a) 1812 (b) 1809 (c) 1813 (d) 1818
26. 26.What is the units digit of 2^25? (a) 6 (b) 2 (c) 4 (d) 8
27. 27.Which of these numbers is divisible by 8? (a) 6957 (b) 6972 (c) 6963 (d) 6960
28. 28.What is the units digit of 7^55? (a) 9 (b) 7 (c) 1 (d) 3
29. 29.What is the remainder when 159 is divided by 7? (a) 4 (b) 2 (c) 6 (d) 5
30. 30.What is the smallest number you must add to 94 to make it divisible by 6? (a) 1 (b) 3 (c) 4 (d) 2
31. 31.Which of these numbers is divisible by 9? (a) 4446 (b) 4439 (c) 4457 (d) 4449
32. 32.What is the remainder when 155 is divided by 11? (a) 0 (b) 10 (c) 2 (d) 1
33. 33.Which of these numbers is divisible by 6? (a) 876 (b) 885 (c) 878 (d) 873
34. 34.What is the remainder when 699 is divided by 10? (a) 7 (b) 1 (c) 8 (d) 9
35. 35.What is the units digit of 3^29? (a) 3 (b) 7 (c) 9 (d) 1
36. 36.What is the units digit of 3^34? (a) 7 (b) 1 (c) 9 (d) 3
37. 37.What is the smallest number you must add to 331 to make it divisible by 10? (a) 9 (b) 7 (c) 1 (d) 8
38. 38.What is the smallest number you must add to 217 to make it divisible by 3? (a) 2 (b) 1
39. 39.Which of these numbers is divisible by 3? (a) 2411 (b) 2413 (c) 2417 (d) 2412
40. 40.Which of these numbers is divisible by 3? (a) 1526 (b) 1531 (c) 1527 (d) 1528

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