Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 511 · 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 8^33? (a) 4 (b) 8 (c) 6 (d) 2
2. 2.What is the remainder when 302 is divided by 7? (a) 1 (b) 6 (c) 0 (d) 2
3. 3.What is the remainder when 112 is divided by 5? (a) 4 (b) 1 (c) 2 (d) 3
4. 4.Which of these numbers is divisible by 3? (a) 1997 (b) 1999 (c) 1995 (d) 1993
5. 5.What is the units digit of 3^39? (a) 7 (b) 9 (c) 1 (d) 3
6. 6.What is the units digit of 2^36? (a) 8 (b) 2 (c) 4 (d) 6
7. 7.Which of these numbers is divisible by 3? (a) 2487 (b) 2488 (c) 2491 (d) 2485
8. 8.Which of these numbers is divisible by 6? (a) 1420 (b) 1425 (c) 1411 (d) 1416
9. 9.What is the units digit of 2^17? (a) 8 (b) 6 (c) 2 (d) 4
10. 10.What is the units digit of 4^12? (a) 5 (b) 7 (c) 6 (d) 4
11. 11.Which of these numbers is divisible by 4? (a) 1998 (b) 1990 (c) 1992 (d) 1993
12. 12.What is the smallest number you must add to 167 to make it divisible by 8? (a) 1 (b) 2 (c) 7 (d) 3
13. 13.What is the units digit of 9^10? (a) 9 (b) 0 (c) 1 (d) 2
14. 14.What is the units digit of 9^17? (a) 9 (b) 1 (c) 8 (d) 7
15. 15.What is the remainder when 590 is divided by 11? (a) 7 (b) 8 (c) 6 (d) 4
16. 16.What is the units digit of 8^58? (a) 2 (b) 8 (c) 6 (d) 4
17. 17.What is the remainder when 391 is divided by 5? (a) 2 (b) 0 (c) 4 (d) 1
18. 18.What is the remainder when 562 is divided by 9? (a) 4 (b) 3 (c) 5 (d) 6
19. 19.Which of these numbers is divisible by 4? (a) 551 (b) 552 (c) 554 (d) 558
20. 20.What is the smallest number you must add to 284 to make it divisible by 7? (a) 4 (b) 2 (c) 5 (d) 3
21. 21.What is the smallest number you must add to 101 to make it divisible by 3? (a) 1 (b) 2
22. 22.Which of these numbers is divisible by 4? (a) 1468 (b) 1469 (c) 1474 (d) 1467
23. 23.Which of these numbers is divisible by 11? (a) 2439 (b) 2442 (c) 2446 (d) 2460
24. 24.What is the smallest number you must add to 103 to make it divisible by 4? (a) 3 (b) 2 (c) 1
25. 25.What is the remainder when 333 is divided by 6? (a) 3 (b) 4 (c) 5 (d) 2
26. 26.What is the units digit of 8^14? (a) 6 (b) 4 (c) 2 (d) 8
27. 27.Which of these numbers is divisible by 4? (a) 2789 (b) 2793 (c) 2788 (d) 2787
28. 28.Which of these numbers is divisible by 9? (a) 3349 (b) 3345 (c) 3339 (d) 3336
29. 29.Which of these numbers is divisible by 6? (a) 336 (b) 340 (c) 334 (d) 343
30. 30.What is the units digit of 3^26? (a) 9 (b) 3 (c) 1 (d) 7
31. 31.What is the smallest number you must add to 68 to make it divisible by 3? (a) 1 (b) 2
32. 32.Which of these numbers is divisible by 11? (a) 9666 (b) 9678 (c) 9648 (d) 9658
33. 33.What is the remainder when 93 is divided by 9? (a) 4 (b) 6 (c) 3 (d) 2
34. 34.Which of these numbers is divisible by 3? (a) 442 (b) 439 (c) 436 (d) 438
35. 35.What is the units digit of 8^43? (a) 2 (b) 4 (c) 6 (d) 8
36. 36.Which of these numbers is divisible by 9? (a) 4529 (b) 4526 (c) 4517 (d) 4518
37. 37.Which of these numbers is divisible by 3? (a) 1003 (b) 1000 (c) 1007 (d) 1002
38. 38.What is the remainder when 442 is divided by 9? (a) 0 (b) 8 (c) 2 (d) 1
39. 39.What is the smallest number you must add to 55 to make it divisible by 4? (a) 2 (b) 3 (c) 1
40. 40.Which of these numbers is divisible by 11? (a) 5851 (b) 5832 (c) 5841 (d) 5862

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