Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 954 · 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 9? (a) 5504 (b) 5510 (c) 5520 (d) 5508
2. 2.Which of these numbers is divisible by 6? (a) 2895 (b) 2888 (c) 2902 (d) 2892
3. 3.What is the smallest number you must add to 421 to make it divisible by 6? (a) 1 (b) 4 (c) 3 (d) 5
4. 4.What is the smallest number you must add to 476 to make it divisible by 6? (a) 5 (b) 4 (c) 2 (d) 3
5. 5.What is the remainder when 114 is divided by 4? (a) 2 (b) 0 (c) 3 (d) 1
6. 6.What is the remainder when 167 is divided by 5? (a) 3 (b) 4 (c) 2 (d) 1
7. 7.What is the smallest number you must add to 129 to make it divisible by 5? (a) 2 (b) 4 (c) 3 (d) 1
8. 8.Which of these numbers is divisible by 4? (a) 574 (b) 572 (c) 577 (d) 570
9. 9.What is the remainder when 903 is divided by 12? (a) 4 (b) 9 (c) 3 (d) 2
10. 10.What is the smallest number you must add to 121 to make it divisible by 3? (a) 1 (b) 2
11. 11.Which of these numbers is divisible by 9? (a) 3187 (b) 3177 (c) 3180 (d) 3171
12. 12.What is the smallest number you must add to 327 to make it divisible by 9? (a) 7 (b) 5 (c) 3 (d) 6
13. 13.Which of these numbers is divisible by 11? (a) 3173 (b) 3192 (c) 3180 (d) 3179
14. 14.What is the units digit of 3^32? (a) 7 (b) 3 (c) 9 (d) 1
15. 15.What is the units digit of 8^45? (a) 4 (b) 2 (c) 8 (d) 6
16. 16.What is the units digit of 9^59? (a) 9 (b) 7 (c) 8 (d) 1
17. 17.Which of these numbers is divisible by 4? (a) 2381 (b) 2376 (c) 2375 (d) 2379
18. 18.Which of these numbers is divisible by 6? (a) 786 (b) 782 (c) 794 (d) 788
19. 19.What is the remainder when 373 is divided by 9? (a) 6 (b) 4 (c) 3 (d) 5
20. 20.What is the smallest number you must add to 522 to make it divisible by 8? (a) 7 (b) 2 (c) 5 (d) 6
21. 21.Which of these numbers is divisible by 6? (a) 3562 (b) 3566 (c) 3564 (d) 3571
22. 22.What is the smallest number you must add to 374 to make it divisible by 7? (a) 4 (b) 5 (c) 3 (d) 6
23. 23.What is the units digit of 9^40? (a) 9 (b) 2 (c) 0 (d) 1
24. 24.What is the units digit of 7^18? (a) 1 (b) 3 (c) 9 (d) 7
25. 25.What is the remainder when 303 is divided by 12? (a) 4 (b) 3 (c) 2 (d) 9
26. 26.What is the smallest number you must add to 179 to make it divisible by 12? (a) 11 (b) 3 (c) 1 (d) 2
27. 27.Which of these numbers is divisible by 3? (a) 1002 (b) 1007 (c) 1001 (d) 1003
28. 28.What is the smallest number you must add to 92 to make it divisible by 6? (a) 5 (b) 4 (c) 2 (d) 3
29. 29.What is the smallest number you must add to 250 to make it divisible by 6? (a) 2 (b) 1 (c) 4 (d) 3
30. 30.What is the units digit of 7^15? (a) 1 (b) 7 (c) 9 (d) 3
31. 31.Which of these numbers is divisible by 3? (a) 2434 (b) 2431 (c) 2433 (d) 2438
32. 32.Which of these numbers is divisible by 8? (a) 4544 (b) 4553 (c) 4550 (d) 4538
33. 33.What is the remainder when 121 is divided by 5? (a) 0 (b) 4 (c) 2 (d) 1
34. 34.Which of these numbers is divisible by 6? (a) 3569 (b) 3577 (c) 3574 (d) 3570
35. 35.Which of these numbers is divisible by 8? (a) 7008 (b) 7006 (c) 7013 (d) 7023
36. 36.What is the remainder when 120 is divided by 7? (a) 6 (b) 0 (c) 2 (d) 1
37. 37.Which of these numbers is divisible by 8? (a) 2831 (b) 2838 (c) 2832 (d) 2843
38. 38.What is the smallest number you must add to 277 to make it divisible by 9? (a) 2 (b) 7 (c) 1 (d) 3
39. 39.What is the remainder when 519 is divided by 7? (a) 6 (b) 2 (c) 0 (d) 1
40. 40.What is the units digit of 2^55? (a) 8 (b) 4 (c) 6 (d) 2

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