Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 686 · 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 743 is divided by 12? (a) 11 (b) 10 (c) 1 (d) 9
2. 2.What is the smallest number you must add to 641 to make it divisible by 8? (a) 5 (b) 7 (c) 6 (d) 1
3. 3.What is the smallest number you must add to 138 to make it divisible by 10? (a) 1 (b) 8 (c) 2 (d) 3
4. 4.What is the units digit of 7^30? (a) 9 (b) 1 (c) 3 (d) 7
5. 5.What is the smallest number you must add to 887 to make it divisible by 11? (a) 5 (b) 7 (c) 3 (d) 4
6. 6.What is the units digit of 8^49? (a) 2 (b) 8 (c) 6 (d) 4
7. 7.Which of these numbers is divisible by 6? (a) 2900 (b) 2904 (c) 2905 (d) 2913
8. 8.What is the units digit of 2^26? (a) 2 (b) 6 (c) 8 (d) 4
9. 9.What is the units digit of 4^59? (a) 6 (b) 4 (c) 5 (d) 3
10. 10.What is the remainder when 397 is divided by 8? (a) 3 (b) 5 (c) 4 (d) 6
11. 11.What is the remainder when 708 is divided by 10? (a) 9 (b) 2 (c) 8 (d) 7
12. 12.What is the smallest number you must add to 235 to make it divisible by 3? (a) 1 (b) 2
13. 13.What is the remainder when 349 is divided by 7? (a) 4 (b) 5 (c) 6 (d) 1
14. 14.What is the smallest number you must add to 371 to make it divisible by 9? (a) 8 (b) 6 (c) 2 (d) 7
15. 15.What is the units digit of 2^59? (a) 6 (b) 2 (c) 4 (d) 8
16. 16.Which of these numbers is divisible by 9? (a) 5982 (b) 5972 (c) 5961 (d) 5967
17. 17.What is the units digit of 4^16? (a) 7 (b) 6 (c) 5 (d) 4
18. 18.What is the units digit of 3^20? (a) 3 (b) 1 (c) 7 (d) 9
19. 19.What is the units digit of 9^34? (a) 0 (b) 9 (c) 1 (d) 2
20. 20.Which of these numbers is divisible by 8? (a) 6169 (b) 6191 (c) 6181 (d) 6176
21. 21.What is the units digit of 8^46? (a) 2 (b) 8 (c) 6 (d) 4
22. 22.Which of these numbers is divisible by 6? (a) 1657 (b) 1673 (c) 1662 (d) 1665
23. 23.What is the units digit of 7^21? (a) 1 (b) 7 (c) 3 (d) 9
24. 24.Which of these numbers is divisible by 4? (a) 792 (b) 793 (c) 798 (d) 790
25. 25.What is the remainder when 361 is divided by 5? (a) 4 (b) 2 (c) 0 (d) 1
26. 26.What is the smallest number you must add to 295 to make it divisible by 4? (a) 3 (b) 2 (c) 1
27. 27.What is the remainder when 535 is divided by 7? (a) 4 (b) 3 (c) 5 (d) 2
28. 28.Which of these numbers is divisible by 6? (a) 2507 (b) 2499 (c) 2502 (d) 2510
29. 29.What is the remainder when 398 is divided by 10? (a) 9 (b) 7 (c) 8 (d) 2
30. 30.What is the remainder when 76 is divided by 5? (a) 4 (b) 2 (c) 0 (d) 1
31. 31.Which of these numbers is divisible by 6? (a) 2133 (b) 2120 (c) 2125 (d) 2124
32. 32.Which of these numbers is divisible by 8? (a) 6333 (b) 6338 (c) 6336 (d) 6345
33. 33.What is the remainder when 249 is divided by 8? (a) 2 (b) 1 (c) 0 (d) 7
34. 34.What is the remainder when 68 is divided by 6? (a) 3 (b) 2 (c) 4 (d) 1
35. 35.What is the remainder when 299 is divided by 4? (a) 2 (b) 1 (c) 3 (d) 0
36. 36.What is the units digit of 8^53? (a) 8 (b) 2 (c) 6 (d) 4
37. 37.Which of these numbers is divisible by 9? (a) 7209 (b) 7215 (c) 7206 (d) 7226
38. 38.What is the smallest number you must add to 683 to make it divisible by 11? (a) 10 (b) 1 (c) 9 (d) 8
39. 39.Which of these numbers is divisible by 11? (a) 5605 (b) 5611 (c) 5591 (d) 5599
40. 40.What is the remainder when 99 is divided by 4? (a) 2 (b) 1 (c) 0 (d) 3

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