Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 700 · 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 9? (a) 5861 (b) 5859 (c) 5851 (d) 5869
2. 2.What is the remainder when 415 is divided by 6? (a) 5 (b) 2 (c) 0 (d) 1
3. 3.What is the remainder when 518 is divided by 11? (a) 0 (b) 10 (c) 2 (d) 1
4. 4.Which of these numbers is divisible by 8? (a) 2469 (b) 2459 (c) 2464 (d) 2476
5. 5.Which of these numbers is divisible by 3? (a) 2071 (b) 2069 (c) 2067 (d) 2065
6. 6.What is the units digit of 4^35? (a) 6 (b) 3 (c) 4 (d) 5
7. 7.What is the smallest number you must add to 516 to make it divisible by 8? (a) 5 (b) 6 (c) 4 (d) 3
8. 8.What is the remainder when 34 is divided by 3? (a) 2 (b) 1 (c) 0
9. 9.What is the remainder when 140 is divided by 8? (a) 6 (b) 4 (c) 5 (d) 3
10. 10.What is the remainder when 392 is divided by 9? (a) 7 (b) 6 (c) 5 (d) 4
11. 11.What is the smallest number you must add to 715 to make it divisible by 9? (a) 6 (b) 5 (c) 4 (d) 7
12. 12.What is the units digit of 7^27? (a) 1 (b) 9 (c) 3 (d) 7
13. 13.Which of these numbers is divisible by 3? (a) 970 (b) 966 (c) 965 (d) 967
14. 14.What is the smallest number you must add to 521 to make it divisible by 8? (a) 7 (b) 1 (c) 6 (d) 5
15. 15.What is the units digit of 4^37? (a) 6 (b) 5 (c) 3 (d) 4
16. 16.What is the remainder when 78 is divided by 4? (a) 1 (b) 3 (c) 0 (d) 2
17. 17.Which of these numbers is divisible by 3? (a) 2171 (b) 2173 (c) 2168 (d) 2169
18. 18.What is the remainder when 172 is divided by 9? (a) 1 (b) 2 (c) 8 (d) 0
19. 19.What is the units digit of 9^21? (a) 8 (b) 7 (c) 1 (d) 9
20. 20.What is the remainder when 484 is divided by 12? (a) 4 (b) 3 (c) 5 (d) 8
21. 21.What is the units digit of 2^44? (a) 6 (b) 2 (c) 8 (d) 4
22. 22.What is the smallest number you must add to 339 to make it divisible by 6? (a) 2 (b) 5 (c) 3 (d) 4
23. 23.What is the units digit of 3^32? (a) 9 (b) 1 (c) 3 (d) 7
24. 24.Which of these numbers is divisible by 8? (a) 1754 (b) 1766 (c) 1747 (d) 1752
25. 25.What is the units digit of 4^57? (a) 3 (b) 5 (c) 4 (d) 6
26. 26.What is the remainder when 601 is divided by 8? (a) 7 (b) 1 (c) 0 (d) 2
27. 27.Which of these numbers is divisible by 4? (a) 299 (b) 293 (c) 292 (d) 291
28. 28.What is the smallest number you must add to 58 to make it divisible by 3? (a) 2 (b) 1
29. 29.What is the units digit of 2^48? (a) 8 (b) 6 (c) 4 (d) 2
30. 30.What is the units digit of 3^57? (a) 7 (b) 9 (c) 1 (d) 3
31. 31.Which of these numbers is divisible by 9? (a) 6352 (b) 6338 (c) 6336 (d) 6330
32. 32.What is the units digit of 3^12? (a) 9 (b) 1 (c) 3 (d) 7
33. 33.What is the units digit of 2^19? (a) 2 (b) 6 (c) 4 (d) 8
34. 34.What is the units digit of 8^28? (a) 4 (b) 8 (c) 6 (d) 2
35. 35.Which of these numbers is divisible by 4? (a) 729 (b) 733 (c) 728 (d) 727
36. 36.What is the smallest number you must add to 398 to make it divisible by 6? (a) 5 (b) 4 (c) 3 (d) 2
37. 37.What is the remainder when 561 is divided by 7? (a) 0 (b) 6 (c) 2 (d) 1
38. 38.What is the remainder when 629 is divided by 12? (a) 4 (b) 6 (c) 7 (d) 5
39. 39.What is the smallest number you must add to 731 to make it divisible by 10? (a) 8 (b) 9 (c) 1 (d) 7
40. 40.Which of these numbers is divisible by 6? (a) 3461 (b) 3446 (c) 3451 (d) 3450

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