Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 765 · 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 smallest number you must add to 535 to make it divisible by 7? (a) 5 (b) 6 (c) 3 (d) 4
2. 2.What is the remainder when 334 is divided by 8? (a) 2 (b) 5 (c) 7 (d) 6
3. 3.What is the remainder when 114 is divided by 8? (a) 6 (b) 2 (c) 1 (d) 3
4. 4.What is the remainder when 158 is divided by 7? (a) 6 (b) 5 (c) 4 (d) 3
5. 5.Which of these numbers is divisible by 11? (a) 8541 (b) 8525 (c) 8516 (d) 8531
6. 6.Which of these numbers is divisible by 11? (a) 3829 (b) 3839 (c) 3855 (d) 3844
7. 7.What is the smallest number you must add to 779 to make it divisible by 11? (a) 3 (b) 9 (c) 1 (d) 2
8. 8.Which of these numbers is divisible by 3? (a) 729 (b) 734 (c) 728 (d) 731
9. 9.Which of these numbers is divisible by 4? (a) 2105 (b) 2103 (c) 2100 (d) 2097
10. 10.What is the units digit of 3^48? (a) 3 (b) 1 (c) 9 (d) 7
11. 11.What is the remainder when 406 is divided by 8? (a) 7 (b) 5 (c) 6 (d) 2
12. 12.What is the smallest number you must add to 238 to make it divisible by 4? (a) 3 (b) 2 (c) 1
13. 13.What is the remainder when 463 is divided by 10? (a) 2 (b) 7 (c) 4 (d) 3
14. 14.Which of these numbers is divisible by 4? (a) 2614 (b) 2616 (c) 2619 (d) 2622
15. 15.What is the remainder when 89 is divided by 5? (a) 2 (b) 3 (c) 4 (d) 1
16. 16.What is the remainder when 370 is divided by 7? (a) 6 (b) 4 (c) 5 (d) 1
17. 17.What is the smallest number you must add to 186 to make it divisible by 11? (a) 10 (b) 1 (c) 3 (d) 2
18. 18.What is the remainder when 363 is divided by 5? (a) 1 (b) 2 (c) 3 (d) 4
19. 19.Which of these numbers is divisible by 9? (a) 4754 (b) 4748 (c) 4737 (d) 4743
20. 20.What is the units digit of 4^14? (a) 4 (b) 6 (c) 7 (d) 5
21. 21.What is the remainder when 301 is divided by 6? (a) 5 (b) 2 (c) 1 (d) 0
22. 22.Which of these numbers is divisible by 6? (a) 1933 (b) 1929 (c) 1932 (d) 1942
23. 23.What is the smallest number you must add to 186 to make it divisible by 4? (a) 3 (b) 2 (c) 1
24. 24.Which of these numbers is divisible by 8? (a) 747 (b) 739 (c) 758 (d) 744
25. 25.What is the remainder when 739 is divided by 10? (a) 7 (b) 9 (c) 1 (d) 8
26. 26.What is the units digit of 9^34? (a) 1 (b) 2 (c) 9 (d) 0
27. 27.What is the remainder when 389 is divided by 5? (a) 4 (b) 2 (c) 1 (d) 3
28. 28.Which of these numbers is divisible by 9? (a) 949 (b) 945 (c) 961 (d) 944
29. 29.What is the units digit of 7^34? (a) 3 (b) 9 (c) 1 (d) 7
30. 30.What is the remainder when 316 is divided by 6? (a) 4 (b) 3 (c) 2 (d) 5
31. 31.What is the remainder when 790 is divided by 12? (a) 11 (b) 10 (c) 9 (d) 2
32. 32.What is the remainder when 189 is divided by 4? (a) 1 (b) 2 (c) 0 (d) 3
33. 33.What is the units digit of 4^45? (a) 4 (b) 3 (c) 5 (d) 6
34. 34.What is the smallest number you must add to 493 to make it divisible by 7? (a) 6 (b) 3 (c) 4 (d) 5
35. 35.What is the smallest number you must add to 339 to make it divisible by 12? (a) 3 (b) 10 (c) 9 (d) 8
36. 36.What is the smallest number you must add to 241 to make it divisible by 5? (a) 3 (b) 1 (c) 2 (d) 4
37. 37.Which of these numbers is divisible by 3? (a) 1116 (b) 1117 (c) 1115 (d) 1120
38. 38.What is the units digit of 7^15? (a) 7 (b) 9 (c) 3 (d) 1
39. 39.Which of these numbers is divisible by 8? (a) 6368 (b) 6382 (c) 6361 (d) 6373
40. 40.Which of these numbers is divisible by 11? (a) 5008 (b) 5016 (c) 5034 (d) 5017

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