Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 106 · 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.What is the remainder when 256 is divided by 9? (a) 5 (b) 6 (c) 4 (d) 3
2. 2.What is the remainder when 640 is divided by 9? (a) 1 (b) 2 (c) 0 (d) 8
3. 3.What is the units digit of 7^38? (a) 3 (b) 1 (c) 7 (d) 9
4. 4.Which of these numbers is divisible by 9? (a) 702 (b) 709 (c) 713 (d) 697
5. 5.What is the smallest number you must add to 162 to make it divisible by 4? (a) 3 (b) 1 (c) 2
6. 6.What is the remainder when 201 is divided by 7? (a) 2 (b) 6 (c) 5 (d) 4
7. 7.What is the smallest number you must add to 100 to make it divisible by 7? (a) 2 (b) 6 (c) 4 (d) 5
8. 8.Which of these numbers is divisible by 11? (a) 8859 (b) 8838 (c) 8847 (d) 8844
9. 9.Which of these numbers is divisible by 6? (a) 276 (b) 286 (c) 279 (d) 272
10. 10.What is the units digit of 7^59? (a) 1 (b) 3 (c) 9 (d) 7
11. 11.What is the smallest number you must add to 340 to make it divisible by 7? (a) 2 (b) 3 (c) 4 (d) 5
12. 12.What is the remainder when 253 is divided by 4? (a) 2 (b) 1 (c) 3 (d) 0
13. 13.Which of these numbers is divisible by 11? (a) 8513 (b) 8503 (c) 8497 (d) 8515
14. 14.What is the smallest number you must add to 347 to make it divisible by 5? (a) 3 (b) 1 (c) 2 (d) 4
15. 15.What is the smallest number you must add to 77 to make it divisible by 6? (a) 2 (b) 3 (c) 1 (d) 5
16. 16.What is the remainder when 104 is divided by 5? (a) 4 (b) 3 (c) 2 (d) 1
17. 17.What is the remainder when 92 is divided by 3? (a) 1 (b) 0 (c) 2
18. 18.What is the remainder when 287 is divided by 6? (a) 5 (b) 4 (c) 3 (d) 1
19. 19.Which of these numbers is divisible by 9? (a) 1106 (b) 1093 (c) 1109 (d) 1098
20. 20.What is the remainder when 713 is divided by 9? (a) 3 (b) 7 (c) 2 (d) 1
21. 21.What is the smallest number you must add to 435 to make it divisible by 11? (a) 4 (b) 7 (c) 5 (d) 6
22. 22.What is the smallest number you must add to 245 to make it divisible by 8? (a) 5 (b) 4 (c) 2 (d) 3
23. 23.What is the units digit of 2^42? (a) 6 (b) 4 (c) 2 (d) 8
24. 24.What is the smallest number you must add to 364 to make it divisible by 12? (a) 7 (b) 4 (c) 8 (d) 9
25. 25.Which of these numbers is divisible by 6? (a) 1170 (b) 1169 (c) 1171 (d) 1178
26. 26.What is the units digit of 9^12? (a) 0 (b) 9 (c) 1 (d) 2
27. 27.Which of these numbers is divisible by 3? (a) 649 (b) 652 (c) 646 (d) 648
28. 28.What is the remainder when 606 is divided by 10? (a) 7 (b) 4 (c) 5 (d) 6
29. 29.Which of these numbers is divisible by 3? (a) 369 (b) 368 (c) 373 (d) 371
30. 30.Which of these numbers is divisible by 11? (a) 2310 (b) 2322 (c) 2306 (d) 2316
31. 31.Which of these numbers is divisible by 3? (a) 1690 (b) 1687 (c) 1686 (d) 1684
32. 32.What is the smallest number you must add to 331 to make it divisible by 7? (a) 2 (b) 6 (c) 4 (d) 5
33. 33.What is the units digit of 7^35? (a) 3 (b) 1 (c) 9 (d) 7
34. 34.What is the smallest number you must add to 391 to make it divisible by 6? (a) 1 (b) 5 (c) 3 (d) 4
35. 35.Which of these numbers is divisible by 11? (a) 8856 (b) 8855 (c) 8854 (d) 8872
36. 36.What is the remainder when 493 is divided by 10? (a) 3 (b) 4 (c) 2 (d) 7
37. 37.What is the units digit of 7^20? (a) 3 (b) 7 (c) 1 (d) 9
38. 38.What is the remainder when 234 is divided by 5? (a) 4 (b) 3 (c) 1 (d) 2
39. 39.What is the smallest number you must add to 264 to make it divisible by 10? (a) 5 (b) 7 (c) 4 (d) 6
40. 40.What is the remainder when 191 is divided by 4? (a) 0 (b) 2 (c) 1 (d) 3

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