Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 887 · 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 7^21? (a) 7 (b) 1 (c) 3 (d) 9
2. 2.What is the remainder when 222 is divided by 11? (a) 1 (b) 2 (c) 9 (d) 3
3. 3.What is the remainder when 278 is divided by 6? (a) 4 (b) 2 (c) 3 (d) 1
4. 4.What is the units digit of 3^28? (a) 7 (b) 1 (c) 3 (d) 9
5. 5.What is the remainder when 77 is divided by 3? (a) 2 (b) 1 (c) 0
6. 6.What is the smallest number you must add to 179 to make it divisible by 7? (a) 2 (b) 3 (c) 5 (d) 4
7. 7.Which of these numbers is divisible by 8? (a) 544 (b) 542 (c) 549 (d) 559
8. 8.Which of these numbers is divisible by 6? (a) 2100 (b) 2097 (c) 2109 (d) 2102
9. 9.Which of these numbers is divisible by 8? (a) 4008 (b) 4007 (c) 4009 (d) 4019
10. 10.Which of these numbers is divisible by 6? (a) 3668 (b) 3673 (c) 3666 (d) 3661
11. 11.What is the remainder when 281 is divided by 5? (a) 4 (b) 2 (c) 0 (d) 1
12. 12.What is the units digit of 3^36? (a) 9 (b) 1 (c) 3 (d) 7
13. 13.Which of these numbers is divisible by 11? (a) 6899 (b) 6897 (c) 6910 (d) 6887
14. 14.Which of these numbers is divisible by 3? (a) 2156 (b) 2162 (c) 2157 (d) 2159
15. 15.Which of these numbers is divisible by 8? (a) 6172 (b) 6154 (c) 6160 (d) 6163
16. 16.What is the units digit of 7^50? (a) 7 (b) 1 (c) 9 (d) 3
17. 17.What is the remainder when 552 is divided by 9? (a) 4 (b) 2 (c) 3 (d) 6
18. 18.What is the smallest number you must add to 204 to make it divisible by 10? (a) 7 (b) 4 (c) 5 (d) 6
19. 19.What is the remainder when 587 is divided by 10? (a) 8 (b) 3 (c) 7 (d) 6
20. 20.Which of these numbers is divisible by 9? (a) 3757 (b) 3744 (c) 3746 (d) 3742
21. 21.What is the units digit of 8^31? (a) 4 (b) 6 (c) 2 (d) 8
22. 22.Which of these numbers is divisible by 11? (a) 8008 (b) 8001 (c) 8014 (d) 8026
23. 23.What is the units digit of 4^14? (a) 5 (b) 7 (c) 6 (d) 4
24. 24.Which of these numbers is divisible by 4? (a) 317 (b) 321 (c) 313 (d) 316
25. 25.What is the smallest number you must add to 503 to make it divisible by 7? (a) 6 (b) 1 (c) 3 (d) 2
26. 26.What is the smallest number you must add to 172 to make it divisible by 6? (a) 3 (b) 4 (c) 2 (d) 1
27. 27.What is the units digit of 3^41? (a) 9 (b) 7 (c) 1 (d) 3
28. 28.What is the smallest number you must add to 127 to make it divisible by 3? (a) 1 (b) 2
29. 29.What is the smallest number you must add to 222 to make it divisible by 4? (a) 3 (b) 1 (c) 2
30. 30.What is the units digit of 2^37? (a) 6 (b) 8 (c) 2 (d) 4
31. 31.What is the units digit of 7^46? (a) 3 (b) 7 (c) 9 (d) 1
32. 32.What is the units digit of 8^10? (a) 6 (b) 2 (c) 4 (d) 8
33. 33.What is the units digit of 2^31? (a) 8 (b) 4 (c) 2 (d) 6
34. 34.What is the units digit of 8^19? (a) 8 (b) 4 (c) 2 (d) 6
35. 35.What is the smallest number you must add to 109 to make it divisible by 3? (a) 2 (b) 1
36. 36.What is the units digit of 8^45? (a) 2 (b) 8 (c) 6 (d) 4
37. 37.What is the smallest number you must add to 349 to make it divisible by 6? (a) 5 (b) 1 (c) 3 (d) 4
38. 38.What is the units digit of 7^31? (a) 7 (b) 1 (c) 3 (d) 9
39. 39.Which of these numbers is divisible by 8? (a) 6789 (b) 6784 (c) 6777 (d) 6793
40. 40.What is the smallest number you must add to 311 to make it divisible by 11? (a) 7 (b) 8 (c) 9 (d) 3

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