Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 384 · 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) 6904 (b) 6916 (c) 6908 (d) 6928
2. 2.What is the units digit of 8^49? (a) 2 (b) 4 (c) 8 (d) 6
3. 3.What is the smallest number you must add to 76 to make it divisible by 7? (a) 3 (b) 1 (c) 6 (d) 2
4. 4.Which of these numbers is divisible by 4? (a) 2095 (b) 2090 (c) 2087 (d) 2088
5. 5.What is the units digit of 8^18? (a) 8 (b) 2 (c) 4 (d) 6
6. 6.What is the remainder when 218 is divided by 5? (a) 3 (b) 4 (c) 2 (d) 1
7. 7.What is the remainder when 695 is divided by 10? (a) 7 (b) 5 (c) 6 (d) 4
8. 8.What is the smallest number you must add to 242 to make it divisible by 3? (a) 2 (b) 1
9. 9.What is the units digit of 8^14? (a) 8 (b) 6 (c) 4 (d) 2
10. 10.Which of these numbers is divisible by 8? (a) 1978 (b) 1998 (c) 1984 (d) 1989
11. 11.What is the smallest number you must add to 56 to make it divisible by 5? (a) 4 (b) 2 (c) 1 (d) 3
12. 12.What is the remainder when 170 is divided by 7? (a) 2 (b) 1 (c) 3 (d) 5
13. 13.What is the units digit of 9^39? (a) 9 (b) 7 (c) 8 (d) 1
14. 14.What is the remainder when 217 is divided by 8? (a) 0 (b) 1 (c) 2 (d) 7
15. 15.What is the remainder when 504 is divided by 11? (a) 10 (b) 9 (c) 2 (d) 8
16. 16.What is the remainder when 71 is divided by 7? (a) 6 (b) 1 (c) 0 (d) 2
17. 17.Which of these numbers is divisible by 6? (a) 2422 (b) 2416 (c) 2426 (d) 2418
18. 18.What is the smallest number you must add to 258 to make it divisible by 9? (a) 2 (b) 4 (c) 6 (d) 3
19. 19.Which of these numbers is divisible by 4? (a) 663 (b) 665 (c) 664 (d) 669
20. 20.What is the smallest number you must add to 119 to make it divisible by 6? (a) 3 (b) 5 (c) 1 (d) 2
21. 21.What is the smallest number you must add to 146 to make it divisible by 5? (a) 1 (b) 4 (c) 3 (d) 2
22. 22.Which of these numbers is divisible by 3? (a) 1384 (b) 1388 (c) 1383 (d) 1382
23. 23.What is the units digit of 4^56? (a) 4 (b) 7 (c) 5 (d) 6
24. 24.What is the units digit of 7^37? (a) 1 (b) 7 (c) 9 (d) 3
25. 25.Which of these numbers is divisible by 6? (a) 431 (b) 437 (c) 422 (d) 426
26. 26.Which of these numbers is divisible by 6? (a) 4409 (b) 4410 (c) 4413 (d) 4417
27. 27.What is the smallest number you must add to 322 to make it divisible by 6? (a) 4 (b) 1 (c) 2 (d) 3
28. 28.What is the remainder when 353 is divided by 5? (a) 4 (b) 2 (c) 1 (d) 3
29. 29.What is the smallest number you must add to 199 to make it divisible by 6? (a) 4 (b) 3 (c) 1 (d) 5
30. 30.What is the remainder when 57 is divided by 4? (a) 0 (b) 2 (c) 1 (d) 3
31. 31.What is the units digit of 4^57? (a) 5 (b) 6 (c) 3 (d) 4
32. 32.What is the remainder when 413 is divided by 9? (a) 6 (b) 8 (c) 7 (d) 1
33. 33.Which of these numbers is divisible by 8? (a) 4673 (b) 4680 (c) 4682 (d) 4692
34. 34.Which of these numbers is divisible by 8? (a) 5192 (b) 5198 (c) 5207 (d) 5186
35. 35.Which of these numbers is divisible by 11? (a) 7701 (b) 7730 (c) 7714 (d) 7711
36. 36.Which of these numbers is divisible by 4? (a) 1355 (b) 1351 (c) 1357 (d) 1352
37. 37.What is the smallest number you must add to 209 to make it divisible by 3? (a) 1 (b) 2
38. 38.What is the units digit of 7^11? (a) 1 (b) 9 (c) 3 (d) 7
39. 39.Which of these numbers is divisible by 8? (a) 2896 (b) 2889 (c) 2902 (d) 2911
40. 40.Which of these numbers is divisible by 11? (a) 6801 (b) 6809 (c) 6813 (d) 6826

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