Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 271 · 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 317 is divided by 5? (a) 3 (b) 2 (c) 4 (d) 1
2. 2.Which of these numbers is divisible by 11? (a) 1651 (b) 1650 (c) 1664 (d) 1640
3. 3.Which of these numbers is divisible by 8? (a) 1805 (b) 1789 (c) 1793 (d) 1792
4. 4.Which of these numbers is divisible by 3? (a) 379 (b) 376 (c) 375 (d) 374
5. 5.What is the units digit of 2^33? (a) 6 (b) 4 (c) 8 (d) 2
6. 6.What is the units digit of 3^59? (a) 1 (b) 7 (c) 9 (d) 3
7. 7.What is the smallest number you must add to 172 to make it divisible by 10? (a) 2 (b) 8 (c) 7 (d) 9
8. 8.What is the remainder when 441 is divided by 11? (a) 1 (b) 10 (c) 2 (d) 0
9. 9.What is the remainder when 197 is divided by 9? (a) 8 (b) 7 (c) 6 (d) 1
10. 10.Which of these numbers is divisible by 8? (a) 1765 (b) 1778 (c) 1772 (d) 1768
11. 11.What is the smallest number you must add to 303 to make it divisible by 12? (a) 8 (b) 10 (c) 9 (d) 3
12. 12.Which of these numbers is divisible by 3? (a) 1213 (b) 1212 (c) 1210 (d) 1216
13. 13.What is the remainder when 208 is divided by 3? (a) 1 (b) 2 (c) 0
14. 14.What is the smallest number you must add to 123 to make it divisible by 10? (a) 6 (b) 3 (c) 7 (d) 8
15. 15.What is the units digit of 7^28? (a) 7 (b) 9 (c) 3 (d) 1
16. 16.What is the smallest number you must add to 157 to make it divisible by 4? (a) 2 (b) 3 (c) 1
17. 17.What is the smallest number you must add to 756 to make it divisible by 10? (a) 3 (b) 5 (c) 4 (d) 6
18. 18.What is the units digit of 7^29? (a) 3 (b) 7 (c) 1 (d) 9
19. 19.Which of these numbers is divisible by 8? (a) 1800 (b) 1813 (c) 1807 (d) 1798
20. 20.Which of these numbers is divisible by 11? (a) 2560 (b) 2564 (c) 2552 (d) 2550
21. 21.Which of these numbers is divisible by 3? (a) 119 (b) 120 (c) 121 (d) 124
22. 22.What is the units digit of 3^24? (a) 9 (b) 3 (c) 7 (d) 1
23. 23.What is the units digit of 9^46? (a) 9 (b) 2 (c) 1 (d) 0
24. 24.Which of these numbers is divisible by 11? (a) 4288 (b) 4278 (c) 4297 (d) 4279
25. 25.What is the smallest number you must add to 102 to make it divisible by 9? (a) 5 (b) 7 (c) 3 (d) 6
26. 26.What is the units digit of 3^19? (a) 9 (b) 1 (c) 7 (d) 3
27. 27.Which of these numbers is divisible by 4? (a) 1812 (b) 1809 (c) 1818 (d) 1813
28. 28.What is the remainder when 305 is divided by 4? (a) 3 (b) 2 (c) 0 (d) 1
29. 29.Which of these numbers is divisible by 6? (a) 4411 (b) 4417 (c) 4416 (d) 4423
30. 30.What is the smallest number you must add to 327 to make it divisible by 10? (a) 7 (b) 2 (c) 4 (d) 3
31. 31.Which of these numbers is divisible by 4? (a) 1383 (b) 1390 (c) 1385 (d) 1384
32. 32.What is the smallest number you must add to 119 to make it divisible by 4? (a) 1 (b) 2 (c) 3
33. 33.Which of these numbers is divisible by 6? (a) 320 (b) 310 (c) 315 (d) 312
34. 34.What is the remainder when 680 is divided by 12? (a) 9 (b) 8 (c) 7 (d) 4
35. 35.Which of these numbers is divisible by 4? (a) 1892 (b) 1894 (c) 1897 (d) 1891
36. 36.What is the remainder when 160 is divided by 6? (a) 2 (b) 5 (c) 4 (d) 3
37. 37.What is the units digit of 7^41? (a) 3 (b) 7 (c) 1 (d) 9
38. 38.What is the remainder when 107 is divided by 9? (a) 1 (b) 8 (c) 6 (d) 7
39. 39.Which of these numbers is divisible by 8? (a) 5920 (b) 5914 (c) 5934 (d) 5927
40. 40.Which of these numbers is divisible by 3? (a) 604 (b) 599 (c) 600 (d) 602

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