Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 273 · 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 units digit of 3^42? (a) 3 (b) 7 (c) 1 (d) 9
2. 2.What is the smallest number you must add to 357 to make it divisible by 11? (a) 6 (b) 7 (c) 5 (d) 8
3. 3.Which of these numbers is divisible by 4? (a) 2370 (b) 2361 (c) 2364 (d) 2366
4. 4.What is the smallest number you must add to 475 to make it divisible by 12? (a) 5 (b) 7 (c) 6 (d) 4
5. 5.What is the smallest number you must add to 366 to make it divisible by 9? (a) 6 (b) 4 (c) 2 (d) 3
6. 6.What is the units digit of 7^48? (a) 7 (b) 3 (c) 9 (d) 1
7. 7.Which of these numbers is divisible by 9? (a) 673 (b) 656 (c) 657 (d) 664
8. 8.Which of these numbers is divisible by 4? (a) 3075 (b) 3068 (c) 3070 (d) 3065
9. 9.What is the remainder when 389 is divided by 5? (a) 4 (b) 3 (c) 1 (d) 2
10. 10.Which of these numbers is divisible by 8? (a) 1011 (b) 1000 (c) 999 (d) 1002
11. 11.What is the smallest number you must add to 615 to make it divisible by 11? (a) 1 (b) 10 (c) 2 (d) 3
12. 12.What is the smallest number you must add to 146 to make it divisible by 5? (a) 1 (b) 2 (c) 3 (d) 4
13. 13.What is the units digit of 7^31? (a) 9 (b) 1 (c) 3 (d) 7
14. 14.What is the remainder when 472 is divided by 11? (a) 8 (b) 10 (c) 9 (d) 1
15. 15.Which of these numbers is divisible by 9? (a) 2303 (b) 2309 (c) 2304 (d) 2321
16. 16.Which of these numbers is divisible by 3? (a) 823 (b) 827 (c) 822 (d) 821
17. 17.What is the remainder when 373 is divided by 7? (a) 2 (b) 1 (c) 5 (d) 3
18. 18.What is the units digit of 9^12? (a) 9 (b) 0 (c) 1 (d) 2
19. 19.What is the remainder when 220 is divided by 12? (a) 8 (b) 5 (c) 3 (d) 4
20. 20.What is the remainder when 118 is divided by 4? (a) 3 (b) 2 (c) 0 (d) 1
21. 21.Which of these numbers is divisible by 9? (a) 385 (b) 368 (c) 369 (d) 376
22. 22.What is the remainder when 67 is divided by 3? (a) 1 (b) 0 (c) 2
23. 23.What is the smallest number you must add to 111 to make it divisible by 8? (a) 1 (b) 7 (c) 2 (d) 3
24. 24.What is the smallest number you must add to 321 to make it divisible by 8? (a) 1 (b) 5 (c) 6 (d) 7
25. 25.Which of these numbers is divisible by 3? (a) 193 (b) 197 (c) 195 (d) 199
26. 26.Which of these numbers is divisible by 3? (a) 1288 (b) 1285 (c) 1284 (d) 1282
27. 27.Which of these numbers is divisible by 8? (a) 2562 (b) 2549 (c) 2556 (d) 2552
28. 28.What is the remainder when 546 is divided by 10? (a) 4 (b) 6 (c) 7 (d) 5
29. 29.Which of these numbers is divisible by 4? (a) 3114 (b) 3111 (c) 3108 (d) 3105
30. 30.Which of these numbers is divisible by 8? (a) 3672 (b) 3687 (c) 3675 (d) 3671
31. 31.Which of these numbers is divisible by 3? (a) 1732 (b) 1736 (c) 1734 (d) 1739
32. 32.Which of these numbers is divisible by 3? (a) 589 (b) 592 (c) 591 (d) 595
33. 33.Which of these numbers is divisible by 6? (a) 1432 (b) 1423 (c) 1439 (d) 1428
34. 34.Which of these numbers is divisible by 8? (a) 5396 (b) 5400 (c) 5409 (d) 5405
35. 35.What is the units digit of 8^14? (a) 4 (b) 2 (c) 6 (d) 8
36. 36.Which of these numbers is divisible by 3? (a) 1741 (b) 1735 (c) 1737 (d) 1738
37. 37.What is the units digit of 3^12? (a) 1 (b) 9 (c) 3 (d) 7
38. 38.What is the remainder when 212 is divided by 7? (a) 1 (b) 2 (c) 5 (d) 3
39. 39.What is the units digit of 2^43? (a) 2 (b) 8 (c) 6 (d) 4
40. 40.What is the remainder when 121 is divided by 6? (a) 5 (b) 2 (c) 0 (d) 1

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