Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 251 · 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 110 is divided by 9? (a) 7 (b) 3 (c) 1 (d) 2
2. 2.Which of these numbers is divisible by 8? (a) 3568 (b) 3569 (c) 3566 (d) 3579
3. 3.What is the remainder when 271 is divided by 6? (a) 5 (b) 0 (c) 2 (d) 1
4. 4.Which of these numbers is divisible by 3? (a) 1950 (b) 1951 (c) 1955 (d) 1949
5. 5.What is the remainder when 218 is divided by 4? (a) 3 (b) 1 (c) 0 (d) 2
6. 6.What is the units digit of 8^16? (a) 4 (b) 8 (c) 2 (d) 6
7. 7.Which of these numbers is divisible by 3? (a) 875 (b) 876 (c) 880 (d) 878
8. 8.What is the remainder when 248 is divided by 12? (a) 9 (b) 4 (c) 7 (d) 8
9. 9.What is the remainder when 78 is divided by 7? (a) 2 (b) 6 (c) 1 (d) 0
10. 10.What is the units digit of 8^45? (a) 2 (b) 4 (c) 8 (d) 6
11. 11.What is the units digit of 9^11? (a) 8 (b) 9 (c) 1 (d) 7
12. 12.What is the smallest number you must add to 699 to make it divisible by 12? (a) 8 (b) 3 (c) 9 (d) 10
13. 13.Which of these numbers is divisible by 9? (a) 5585 (b) 5589 (c) 5603 (d) 5594
14. 14.What is the units digit of 4^57? (a) 5 (b) 4 (c) 3 (d) 6
15. 15.What is the units digit of 4^38? (a) 6 (b) 5 (c) 4 (d) 7
16. 16.What is the remainder when 469 is divided by 6? (a) 0 (b) 1 (c) 5 (d) 2
17. 17.Which of these numbers is divisible by 3? (a) 2083 (b) 2081 (c) 2077 (d) 2079
18. 18.What is the smallest number you must add to 698 to make it divisible by 10? (a) 2 (b) 8 (c) 3 (d) 1
19. 19.What is the smallest number you must add to 109 to make it divisible by 3? (a) 1 (b) 2
20. 20.What is the remainder when 245 is divided by 8? (a) 6 (b) 5 (c) 3 (d) 4
21. 21.What is the units digit of 9^17? (a) 9 (b) 7 (c) 8 (d) 1
22. 22.What is the units digit of 8^51? (a) 4 (b) 6 (c) 8 (d) 2
23. 23.What is the units digit of 9^49? (a) 1 (b) 8 (c) 7 (d) 9
24. 24.What is the remainder when 371 is divided by 6? (a) 4 (b) 5 (c) 3 (d) 1
25. 25.Which of these numbers is divisible by 4? (a) 1498 (b) 1490 (c) 1492 (d) 1495
26. 26.What is the units digit of 7^42? (a) 9 (b) 3 (c) 7 (d) 1
27. 27.What is the remainder when 502 is divided by 10? (a) 2 (b) 1 (c) 3 (d) 8
28. 28.Which of these numbers is divisible by 3? (a) 2325 (b) 2329 (c) 2324 (d) 2326
29. 29.What is the remainder when 247 is divided by 7? (a) 3 (b) 5 (c) 2 (d) 1
30. 30.What is the remainder when 866 is divided by 11? (a) 9 (b) 3 (c) 7 (d) 8
31. 31.Which of these numbers is divisible by 3? (a) 2220 (b) 2219 (c) 2221 (d) 2225
32. 32.What is the smallest number you must add to 114 to make it divisible by 7? (a) 6 (b) 4 (c) 5 (d) 2
33. 33.Which of these numbers is divisible by 6? (a) 5161 (b) 5173 (c) 5170 (d) 5166
34. 34.What is the remainder when 130 is divided by 8? (a) 1 (b) 6 (c) 2 (d) 3
35. 35.What is the remainder when 172 is divided by 3? (a) 0 (b) 1 (c) 2
36. 36.What is the remainder when 271 is divided by 7? (a) 6 (b) 5 (c) 2 (d) 4
37. 37.What is the remainder when 519 is divided by 8? (a) 6 (b) 1 (c) 5 (d) 7
38. 38.What is the units digit of 3^18? (a) 9 (b) 3 (c) 7 (d) 1
39. 39.What is the units digit of 3^59? (a) 9 (b) 3 (c) 7 (d) 1
40. 40.What is the smallest number you must add to 85 to make it divisible by 7? (a) 5 (b) 6 (c) 4 (d) 1

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