Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 253 · 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 4? (a) 3367 (b) 3364 (c) 3362 (d) 3371
2. 2.What is the remainder when 381 is divided by 5? (a) 0 (b) 4 (c) 1 (d) 2
3. 3.What is the units digit of 4^11? (a) 5 (b) 3 (c) 4 (d) 6
4. 4.What is the units digit of 7^17? (a) 1 (b) 3 (c) 7 (d) 9
5. 5.Which of these numbers is divisible by 3? (a) 1390 (b) 1393 (c) 1396 (d) 1392
6. 6.What is the remainder when 374 is divided by 5? (a) 2 (b) 3 (c) 4 (d) 1
7. 7.Which of these numbers is divisible by 8? (a) 3113 (b) 3104 (c) 3098 (d) 3105
8. 8.Which of these numbers is divisible by 4? (a) 2511 (b) 2503 (c) 2504 (d) 2505
9. 9.What is the smallest number you must add to 163 to make it divisible by 6? (a) 5 (b) 4 (c) 3 (d) 1
10. 10.What is the units digit of 2^22? (a) 6 (b) 8 (c) 2 (d) 4
11. 11.Which of these numbers is divisible by 6? (a) 4456 (b) 4451 (c) 4463 (d) 4452
12. 12.What is the smallest number you must add to 296 to make it divisible by 12? (a) 5 (b) 8 (c) 3 (d) 4
13. 13.What is the smallest number you must add to 693 to make it divisible by 12? (a) 4 (b) 9 (c) 3 (d) 2
14. 14.Which of these numbers is divisible by 11? (a) 3981 (b) 3988 (c) 4000 (d) 3982
15. 15.What is the smallest number you must add to 314 to make it divisible by 12? (a) 2 (b) 9 (c) 11 (d) 10
16. 16.What is the remainder when 55 is divided by 3? (a) 2 (b) 1 (c) 0
17. 17.What is the units digit of 8^53? (a) 8 (b) 6 (c) 4 (d) 2
18. 18.What is the units digit of 7^25? (a) 3 (b) 9 (c) 7 (d) 1
19. 19.What is the units digit of 7^11? (a) 7 (b) 1 (c) 9 (d) 3
20. 20.What is the remainder when 209 is divided by 8? (a) 7 (b) 2 (c) 0 (d) 1
21. 21.What is the units digit of 9^12? (a) 9 (b) 0 (c) 2 (d) 1
22. 22.What is the units digit of 8^19? (a) 4 (b) 6 (c) 8 (d) 2
23. 23.What is the smallest number you must add to 368 to make it divisible by 5? (a) 3 (b) 4 (c) 2 (d) 1
24. 24.What is the smallest number you must add to 225 to make it divisible by 4? (a) 2 (b) 3 (c) 1
25. 25.What is the smallest number you must add to 68 to make it divisible by 3? (a) 2 (b) 1
26. 26.What is the units digit of 3^41? (a) 7 (b) 9 (c) 1 (d) 3
27. 27.What is the smallest number you must add to 232 to make it divisible by 9? (a) 1 (b) 3 (c) 2 (d) 7
28. 28.Which of these numbers is divisible by 6? (a) 845 (b) 846 (c) 857 (d) 847
29. 29.Which of these numbers is divisible by 3? (a) 2357 (b) 2352 (c) 2353 (d) 2351
30. 30.What is the smallest number you must add to 854 to make it divisible by 12? (a) 11 (b) 10 (c) 9 (d) 2
31. 31.What is the smallest number you must add to 474 to make it divisible by 11? (a) 9 (b) 8 (c) 1 (d) 10
32. 32.What is the remainder when 673 is divided by 9? (a) 7 (b) 2 (c) 8 (d) 6
33. 33.What is the smallest number you must add to 199 to make it divisible by 3? (a) 1 (b) 2
34. 34.What is the smallest number you must add to 715 to make it divisible by 12? (a) 7 (b) 5 (c) 6 (d) 4
35. 35.What is the smallest number you must add to 297 to make it divisible by 6? (a) 3 (b) 5 (c) 4 (d) 2
36. 36.Which of these numbers is divisible by 9? (a) 1156 (b) 1162 (c) 1152 (d) 1148
37. 37.What is the remainder when 311 is divided by 11? (a) 2 (b) 4 (c) 3 (d) 8
38. 38.Which of these numbers is divisible by 11? (a) 5898 (b) 5891 (c) 5885 (d) 5877
39. 39.What is the units digit of 4^53? (a) 6 (b) 3 (c) 4 (d) 5
40. 40.What is the remainder when 224 is divided by 6? (a) 4 (b) 1 (c) 3 (d) 2

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