Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 211 · 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 11? (a) 6112 (b) 6096 (c) 6105 (d) 6118
2. 2.What is the units digit of 9^21? (a) 8 (b) 7 (c) 1 (d) 9
3. 3.Which of these numbers is divisible by 8? (a) 6203 (b) 6188 (c) 6192 (d) 6197
4. 4.What is the units digit of 8^19? (a) 2 (b) 6 (c) 4 (d) 8
5. 5.What is the remainder when 175 is divided by 3? (a) 0 (b) 1 (c) 2
6. 6.Which of these numbers is divisible by 11? (a) 988 (b) 996 (c) 979 (d) 975
7. 7.What is the smallest number you must add to 426 to make it divisible by 11? (a) 3 (b) 2 (c) 8 (d) 4
8. 8.What is the units digit of 8^13? (a) 4 (b) 2 (c) 6 (d) 8
9. 9.Which of these numbers is divisible by 11? (a) 4164 (b) 4177 (c) 4155 (d) 4158
10. 10.What is the remainder when 386 is divided by 6? (a) 4 (b) 2 (c) 3 (d) 1
11. 11.What is the smallest number you must add to 712 to make it divisible by 9? (a) 1 (b) 8 (c) 7 (d) 6
12. 12.What is the smallest number you must add to 193 to make it divisible by 12? (a) 9 (b) 10 (c) 11 (d) 1
13. 13.What is the smallest number you must add to 62 to make it divisible by 4? (a) 1 (b) 3 (c) 2
14. 14.What is the smallest number you must add to 105 to make it divisible by 4? (a) 1 (b) 3 (c) 2
15. 15.Which of these numbers is divisible by 11? (a) 4782 (b) 4795 (c) 4804 (d) 4785
16. 16.Which of these numbers is divisible by 4? (a) 1248 (b) 1255 (c) 1249 (d) 1247
17. 17.What is the smallest number you must add to 577 to make it divisible by 12? (a) 1 (b) 10 (c) 9 (d) 11
18. 18.What is the remainder when 481 is divided by 7? (a) 5 (b) 6 (c) 2 (d) 4
19. 19.What is the remainder when 449 is divided by 11? (a) 2 (b) 10 (c) 8 (d) 9
20. 20.What is the smallest number you must add to 354 to make it divisible by 8? (a) 7 (b) 6 (c) 5 (d) 2
21. 21.Which of these numbers is divisible by 9? (a) 3213 (b) 3211 (c) 3223 (d) 3219
22. 22.What is the smallest number you must add to 184 to make it divisible by 10? (a) 7 (b) 6 (c) 5 (d) 4
23. 23.What is the remainder when 259 is divided by 12? (a) 5 (b) 6 (c) 7 (d) 8
24. 24.Which of these numbers is divisible by 3? (a) 1479 (b) 1481 (c) 1483 (d) 1477
25. 25.What is the smallest number you must add to 635 to make it divisible by 8? (a) 5 (b) 3 (c) 4 (d) 6
26. 26.What is the units digit of 3^27? (a) 9 (b) 1 (c) 7 (d) 3
27. 27.What is the remainder when 463 is divided by 12? (a) 5 (b) 6 (c) 7 (d) 8
28. 28.What is the units digit of 2^27? (a) 6 (b) 4 (c) 2 (d) 8
29. 29.What is the units digit of 8^11? (a) 2 (b) 4 (c) 8 (d) 6
30. 30.What is the smallest number you must add to 184 to make it divisible by 3? (a) 2 (b) 1
31. 31.What is the units digit of 8^34? (a) 2 (b) 8 (c) 4 (d) 6
32. 32.Which of these numbers is divisible by 4? (a) 2790 (b) 2795 (c) 2797 (d) 2792
33. 33.What is the units digit of 8^10? (a) 8 (b) 2 (c) 4 (d) 6
34. 34.What is the remainder when 193 is divided by 4? (a) 0 (b) 1 (c) 2 (d) 3
35. 35.What is the remainder when 65 is divided by 3? (a) 0 (b) 2 (c) 1
36. 36.Which of these numbers is divisible by 4? (a) 1117 (b) 1120 (c) 1122 (d) 1127
37. 37.Which of these numbers is divisible by 4? (a) 371 (b) 374 (c) 368 (d) 367
38. 38.Which of these numbers is divisible by 3? (a) 1519 (b) 1525 (c) 1521 (d) 1522
39. 39.Which of these numbers is divisible by 6? (a) 385 (b) 381 (c) 392 (d) 384
40. 40.Which of these numbers is divisible by 8? (a) 5397 (b) 5384 (c) 5382 (d) 5386

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