Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 143 · 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 6? (a) 666 (b) 661 (c) 673 (d) 670
2. 2.What is the remainder when 442 is divided by 10? (a) 2 (b) 3 (c) 8 (d) 1
3. 3.Which of these numbers is divisible by 3? (a) 497 (b) 499 (c) 498 (d) 502
4. 4.Which of these numbers is divisible by 9? (a) 5523 (b) 5526 (c) 5542 (d) 5534
5. 5.Which of these numbers is divisible by 8? (a) 4444 (b) 4435 (c) 4432 (d) 4430
6. 6.What is the remainder when 403 is divided by 8? (a) 5 (b) 2 (c) 4 (d) 3
7. 7.What is the remainder when 102 is divided by 5? (a) 2 (b) 3 (c) 4 (d) 1
8. 8.Which of these numbers is divisible by 3? (a) 664 (b) 667 (c) 670 (d) 666
9. 9.What is the remainder when 554 is divided by 10? (a) 3 (b) 5 (c) 4 (d) 6
10. 10.What is the remainder when 197 is divided by 8? (a) 3 (b) 4 (c) 6 (d) 5
11. 11.Which of these numbers is divisible by 11? (a) 2228 (b) 2233 (c) 2238 (d) 2252
12. 12.Which of these numbers is divisible by 3? (a) 522 (b) 524 (c) 527 (d) 520
13. 13.What is the units digit of 4^29? (a) 5 (b) 6 (c) 4 (d) 3
14. 14.What is the smallest number you must add to 83 to make it divisible by 8? (a) 3 (b) 4 (c) 5 (d) 6
15. 15.What is the remainder when 258 is divided by 4? (a) 1 (b) 3 (c) 0 (d) 2
16. 16.What is the remainder when 705 is divided by 11? (a) 0 (b) 1 (c) 2 (d) 10
17. 17.What is the remainder when 182 is divided by 9? (a) 1 (b) 2 (c) 3 (d) 7
18. 18.What is the remainder when 377 is divided by 5? (a) 3 (b) 4 (c) 1 (d) 2
19. 19.What is the remainder when 431 is divided by 10? (a) 9 (b) 1 (c) 0 (d) 2
20. 20.What is the smallest number you must add to 131 to make it divisible by 4? (a) 2 (b) 3 (c) 1
21. 21.Which of these numbers is divisible by 4? (a) 1558 (b) 1555 (c) 1561 (d) 1556
22. 22.What is the remainder when 786 is divided by 11? (a) 6 (b) 4 (c) 7 (d) 5
23. 23.Which of these numbers is divisible by 8? (a) 4834 (b) 4826 (c) 4821 (d) 4824
24. 24.Which of these numbers is divisible by 11? (a) 2057 (b) 2048 (c) 2060 (d) 2069
25. 25.Which of these numbers is divisible by 8? (a) 1173 (b) 1168 (c) 1179 (d) 1165
26. 26.What is the units digit of 7^39? (a) 9 (b) 7 (c) 3 (d) 1
27. 27.Which of these numbers is divisible by 4? (a) 1546 (b) 1541 (c) 1544 (d) 1550
28. 28.What is the units digit of 4^37? (a) 6 (b) 4 (c) 3 (d) 5
29. 29.What is the smallest number you must add to 583 to make it divisible by 8? (a) 2 (b) 3 (c) 1 (d) 7
30. 30.What is the remainder when 245 is divided by 4? (a) 3 (b) 0 (c) 2 (d) 1
31. 31.What is the units digit of 3^22? (a) 1 (b) 7 (c) 9 (d) 3
32. 32.What is the smallest number you must add to 113 to make it divisible by 4? (a) 1 (b) 2 (c) 3
33. 33.Which of these numbers is divisible by 6? (a) 538 (b) 545 (c) 549 (d) 540
34. 34.Which of these numbers is divisible by 6? (a) 1011 (b) 1017 (c) 1021 (d) 1014
35. 35.What is the units digit of 2^42? (a) 2 (b) 8 (c) 6 (d) 4
36. 36.What is the smallest number you must add to 427 to make it divisible by 6? (a) 1 (b) 3 (c) 4 (d) 5
37. 37.What is the remainder when 517 is divided by 7? (a) 4 (b) 5 (c) 6 (d) 1
38. 38.What is the smallest number you must add to 64 to make it divisible by 6? (a) 1 (b) 4 (c) 3 (d) 2
39. 39.What is the remainder when 653 is divided by 9? (a) 6 (b) 5 (c) 4 (d) 7
40. 40.What is the smallest number you must add to 221 to make it divisible by 4? (a) 3 (b) 2 (c) 1

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