Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 544 · 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 282 is divided by 5? (a) 3 (b) 4 (c) 1 (d) 2
2. 2.What is the units digit of 7^28? (a) 1 (b) 9 (c) 3 (d) 7
3. 3.What is the units digit of 9^38? (a) 9 (b) 2 (c) 0 (d) 1
4. 4.Which of these numbers is divisible by 11? (a) 4345 (b) 4359 (c) 4354 (d) 4341
5. 5.What is the smallest number you must add to 488 to make it divisible by 11? (a) 6 (b) 7 (c) 8 (d) 4
6. 6.What is the units digit of 3^13? (a) 3 (b) 9 (c) 7 (d) 1
7. 7.What is the remainder when 686 is divided by 9? (a) 2 (b) 3 (c) 1 (d) 7
8. 8.What is the smallest number you must add to 193 to make it divisible by 9? (a) 6 (b) 4 (c) 7 (d) 5
9. 9.Which of these numbers is divisible by 8? (a) 2066 (b) 2073 (c) 2072 (d) 2086
10. 10.Which of these numbers is divisible by 4? (a) 2107 (b) 2102 (c) 2099 (d) 2100
11. 11.What is the smallest number you must add to 433 to make it divisible by 8? (a) 1 (b) 7 (c) 6 (d) 5
12. 12.What is the remainder when 277 is divided by 5? (a) 1 (b) 3 (c) 4 (d) 2
13. 13.Which of these numbers is divisible by 6? (a) 1526 (b) 1537 (c) 1530 (d) 1534
14. 14.Which of these numbers is divisible by 9? (a) 6506 (b) 6494 (c) 6510 (d) 6498
15. 15.What is the smallest number you must add to 396 to make it divisible by 5? (a) 2 (b) 1 (c) 3 (d) 4
16. 16.What is the remainder when 225 is divided by 10? (a) 6 (b) 5 (c) 4 (d) 7
17. 17.What is the remainder when 519 is divided by 7? (a) 2 (b) 0 (c) 6 (d) 1
18. 18.What is the smallest number you must add to 263 to make it divisible by 9? (a) 7 (b) 6 (c) 8 (d) 2
19. 19.Which of these numbers is divisible by 6? (a) 3356 (b) 3354 (c) 3350 (d) 3362
20. 20.What is the units digit of 4^42? (a) 7 (b) 5 (c) 6 (d) 4
21. 21.Which of these numbers is divisible by 3? (a) 842 (b) 847 (c) 843 (d) 844
22. 22.What is the smallest number you must add to 152 to make it divisible by 3? (a) 2 (b) 1
23. 23.What is the units digit of 8^18? (a) 8 (b) 2 (c) 6 (d) 4
24. 24.What is the units digit of 9^19? (a) 9 (b) 1 (c) 7 (d) 8
25. 25.What is the smallest number you must add to 64 to make it divisible by 3? (a) 2 (b) 1
26. 26.Which of these numbers is divisible by 4? (a) 1519 (b) 1516 (c) 1522 (d) 1514
27. 27.What is the remainder when 259 is divided by 6? (a) 0 (b) 5 (c) 2 (d) 1
28. 28.What is the remainder when 342 is divided by 5? (a) 3 (b) 1 (c) 4 (d) 2
29. 29.What is the remainder when 472 is divided by 11? (a) 8 (b) 9 (c) 1 (d) 10
30. 30.What is the smallest number you must add to 92 to make it divisible by 8? (a) 4 (b) 3 (c) 5 (d) 6
31. 31.What is the units digit of 4^24? (a) 5 (b) 7 (c) 4 (d) 6
32. 32.What is the smallest number you must add to 71 to make it divisible by 5? (a) 3 (b) 2 (c) 1 (d) 4
33. 33.What is the remainder when 271 is divided by 5? (a) 0 (b) 1 (c) 4 (d) 2
34. 34.What is the remainder when 81 is divided by 4? (a) 0 (b) 2 (c) 3 (d) 1
35. 35.What is the smallest number you must add to 796 to make it divisible by 12? (a) 9 (b) 8 (c) 7 (d) 4
36. 36.What is the smallest number you must add to 451 to make it divisible by 9? (a) 6 (b) 7 (c) 1 (d) 8
37. 37.What is the smallest number you must add to 537 to make it divisible by 10? (a) 3 (b) 4 (c) 2 (d) 7
38. 38.Which of these numbers is divisible by 6? (a) 1373 (b) 1385 (c) 1375 (d) 1374
39. 39.What is the smallest number you must add to 544 to make it divisible by 9? (a) 7 (b) 5 (c) 4 (d) 6
40. 40.What is the units digit of 2^18? (a) 2 (b) 4 (c) 8 (d) 6

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