Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 692 · 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) 4997 (b) 5002 (c) 5007 (d) 4998
2. 2.What is the remainder when 316 is divided by 9? (a) 2 (b) 1 (c) 8 (d) 0
3. 3.What is the smallest number you must add to 86 to make it divisible by 5? (a) 1 (b) 3 (c) 4 (d) 2
4. 4.What is the units digit of 2^25? (a) 4 (b) 8 (c) 6 (d) 2
5. 5.What is the smallest number you must add to 503 to make it divisible by 12? (a) 11 (b) 3 (c) 2 (d) 1
6. 6.Which of these numbers is divisible by 6? (a) 925 (b) 931 (c) 924 (d) 922
7. 7.What is the remainder when 366 is divided by 5? (a) 0 (b) 4 (c) 2 (d) 1
8. 8.What is the remainder when 760 is divided by 12? (a) 8 (b) 5 (c) 4 (d) 3
9. 9.What is the remainder when 347 is divided by 8? (a) 2 (b) 4 (c) 5 (d) 3
10. 10.What is the smallest number you must add to 103 to make it divisible by 9? (a) 7 (b) 4 (c) 5 (d) 6
11. 11.What is the units digit of 2^27? (a) 6 (b) 8 (c) 4 (d) 2
12. 12.What is the remainder when 373 is divided by 7? (a) 2 (b) 3 (c) 5 (d) 1
13. 13.Which of these numbers is divisible by 6? (a) 3905 (b) 3913 (c) 3906 (d) 3908
14. 14.Which of these numbers is divisible by 9? (a) 6896 (b) 6885 (c) 6878 (d) 6891
15. 15.What is the remainder when 169 is divided by 12? (a) 0 (b) 1 (c) 11 (d) 2
16. 16.Which of these numbers is divisible by 8? (a) 1157 (b) 1139 (c) 1150 (d) 1144
17. 17.What is the remainder when 151 is divided by 11? (a) 8 (b) 3 (c) 9 (d) 7
18. 18.Which of these numbers is divisible by 11? (a) 5363 (b) 5377 (c) 5347 (d) 5357
19. 19.What is the units digit of 2^35? (a) 2 (b) 6 (c) 4 (d) 8
20. 20.What is the remainder when 168 is divided by 5? (a) 2 (b) 3 (c) 1 (d) 4
21. 21.What is the smallest number you must add to 223 to make it divisible by 7? (a) 6 (b) 2 (c) 3 (d) 1
22. 22.What is the smallest number you must add to 182 to make it divisible by 4? (a) 3 (b) 2 (c) 1
23. 23.What is the units digit of 9^33? (a) 9 (b) 8 (c) 7 (d) 1
24. 24.Which of these numbers is divisible by 4? (a) 751 (b) 744 (c) 741 (d) 747
25. 25.What is the units digit of 9^10? (a) 2 (b) 9 (c) 0 (d) 1
26. 26.What is the remainder when 111 is divided by 5? (a) 0 (b) 4 (c) 1 (d) 2
27. 27.What is the smallest number you must add to 127 to make it divisible by 5? (a) 4 (b) 2 (c) 1 (d) 3
28. 28.Which of these numbers is divisible by 3? (a) 506 (b) 500 (c) 501 (d) 502
29. 29.Which of these numbers is divisible by 8? (a) 4793 (b) 4803 (c) 4792 (d) 4787
30. 30.Which of these numbers is divisible by 6? (a) 1367 (b) 1368 (c) 1378 (d) 1371
31. 31.What is the smallest number you must add to 79 to make it divisible by 5? (a) 1 (b) 3 (c) 2 (d) 4
32. 32.What is the smallest number you must add to 385 to make it divisible by 10? (a) 4 (b) 6 (c) 7 (d) 5
33. 33.What is the units digit of 8^38? (a) 4 (b) 8 (c) 2 (d) 6
34. 34.What is the units digit of 3^11? (a) 7 (b) 9 (c) 3 (d) 1
35. 35.What is the remainder when 352 is divided by 5? (a) 1 (b) 4 (c) 2 (d) 3
36. 36.What is the units digit of 2^47? (a) 4 (b) 2 (c) 8 (d) 6
37. 37.What is the remainder when 610 is divided by 12? (a) 9 (b) 11 (c) 2 (d) 10
38. 38.What is the remainder when 85 is divided by 4? (a) 3 (b) 2 (c) 1 (d) 0
39. 39.What is the remainder when 134 is divided by 3? (a) 2 (b) 0 (c) 1
40. 40.Which of these numbers is divisible by 9? (a) 4498 (b) 4491 (c) 4504 (d) 4490

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