Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 962 · 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.Which of these numbers is divisible by 8? (a) 6111 (b) 6097 (c) 6092 (d) 6096
2. 2.What is the remainder when 235 is divided by 3? (a) 2 (b) 0 (c) 1
3. 3.What is the smallest number you must add to 204 to make it divisible by 10? (a) 6 (b) 4 (c) 5 (d) 7
4. 4.What is the remainder when 520 is divided by 9? (a) 2 (b) 8 (c) 6 (d) 7
5. 5.What is the remainder when 138 is divided by 5? (a) 4 (b) 2 (c) 1 (d) 3
6. 6.What is the units digit of 9^35? (a) 7 (b) 1 (c) 8 (d) 9
7. 7.What is the smallest number you must add to 380 to make it divisible by 8? (a) 3 (b) 5 (c) 4 (d) 6
8. 8.What is the smallest number you must add to 229 to make it divisible by 3? (a) 1 (b) 2
9. 9.Which of these numbers is divisible by 9? (a) 4869 (b) 4863 (c) 4881 (d) 4875
10. 10.What is the remainder when 225 is divided by 10? (a) 5 (b) 4 (c) 6 (d) 7
11. 11.Which of these numbers is divisible by 4? (a) 1257 (b) 1252 (c) 1254 (d) 1249
12. 12.What is the units digit of 9^14? (a) 1 (b) 9 (c) 0 (d) 2
13. 13.What is the smallest number you must add to 225 to make it divisible by 4? (a) 1 (b) 3 (c) 2
14. 14.Which of these numbers is divisible by 6? (a) 2473 (b) 2483 (c) 2467 (d) 2472
15. 15.What is the remainder when 215 is divided by 3? (a) 1 (b) 0 (c) 2
16. 16.What is the remainder when 712 is divided by 12? (a) 5 (b) 8 (c) 3 (d) 4
17. 17.What is the units digit of 2^22? (a) 8 (b) 6 (c) 4 (d) 2
18. 18.Which of these numbers is divisible by 11? (a) 2172 (b) 2167 (c) 2161 (d) 2181
19. 19.What is the remainder when 285 is divided by 6? (a) 5 (b) 4 (c) 3 (d) 2
20. 20.What is the smallest number you must add to 359 to make it divisible by 6? (a) 3 (b) 2 (c) 5 (d) 1
21. 21.Which of these numbers is divisible by 11? (a) 8260 (b) 8250 (c) 8249 (d) 8269
22. 22.What is the smallest number you must add to 275 to make it divisible by 7? (a) 5 (b) 4 (c) 2 (d) 6
23. 23.What is the remainder when 83 is divided by 4? (a) 2 (b) 0 (c) 3 (d) 1
24. 24.Which of these numbers is divisible by 3? (a) 1439 (b) 1437 (c) 1435 (d) 1441
25. 25.Which of these numbers is divisible by 11? (a) 3829 (b) 3839 (c) 3856 (d) 3841
26. 26.What is the units digit of 2^30? (a) 6 (b) 2 (c) 8 (d) 4
27. 27.What is the units digit of 8^51? (a) 8 (b) 6 (c) 4 (d) 2
28. 28.Which of these numbers is divisible by 4? (a) 1058 (b) 1061 (c) 1056 (d) 1054
29. 29.What is the remainder when 139 is divided by 4? (a) 1 (b) 2 (c) 0 (d) 3
30. 30.What is the remainder when 877 is divided by 11? (a) 3 (b) 7 (c) 9 (d) 8
31. 31.What is the units digit of 8^19? (a) 4 (b) 8 (c) 6 (d) 2
32. 32.What is the smallest number you must add to 137 to make it divisible by 9? (a) 7 (b) 6 (c) 8 (d) 2
33. 33.What is the remainder when 528 is divided by 7? (a) 5 (b) 2 (c) 4 (d) 3
34. 34.Which of these numbers is divisible by 11? (a) 1841 (b) 1837 (c) 1853 (d) 1830
35. 35.Which of these numbers is divisible by 8? (a) 5366 (b) 5369 (c) 5357 (d) 5360
36. 36.What is the smallest number you must add to 50 to make it divisible by 3? (a) 2 (b) 1
37. 37.What is the remainder when 510 is divided by 11? (a) 5 (b) 3 (c) 4 (d) 7
38. 38.What is the smallest number you must add to 514 to make it divisible by 9? (a) 1 (b) 8 (c) 6 (d) 7
39. 39.Which of these numbers is divisible by 8? (a) 695 (b) 683 (c) 677 (d) 680
40. 40.What is the units digit of 2^32? (a) 2 (b) 6 (c) 8 (d) 4

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