Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 861 · 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 242 is divided by 3? (a) 0 (b) 2 (c) 1
2. 2.What is the units digit of 8^29? (a) 8 (b) 2 (c) 6 (d) 4
3. 3.What is the smallest number you must add to 753 to make it divisible by 11? (a) 8 (b) 6 (c) 5 (d) 7
4. 4.What is the units digit of 4^46? (a) 6 (b) 5 (c) 7 (d) 4
5. 5.Which of these numbers is divisible by 6? (a) 3544 (b) 3551 (c) 3546 (d) 3557
6. 6.What is the remainder when 126 is divided by 4? (a) 3 (b) 0 (c) 2 (d) 1
7. 7.What is the units digit of 7^25? (a) 3 (b) 7 (c) 1 (d) 9
8. 8.What is the units digit of 3^54? (a) 1 (b) 7 (c) 3 (d) 9
9. 9.What is the remainder when 83 is divided by 4? (a) 3 (b) 0 (c) 2 (d) 1
10. 10.What is the units digit of 9^41? (a) 7 (b) 1 (c) 8 (d) 9
11. 11.What is the remainder when 361 is divided by 6? (a) 2 (b) 5 (c) 1 (d) 0
12. 12.What is the units digit of 3^22? (a) 9 (b) 1 (c) 3 (d) 7
13. 13.What is the smallest number you must add to 273 to make it divisible by 6? (a) 4 (b) 5 (c) 3 (d) 2
14. 14.What is the smallest number you must add to 430 to make it divisible by 7? (a) 3 (b) 5 (c) 6 (d) 4
15. 15.What is the smallest number you must add to 31 to make it divisible by 3? (a) 1 (b) 2
16. 16.Which of these numbers is divisible by 9? (a) 494 (b) 509 (c) 499 (d) 495
17. 17.What is the smallest number you must add to 330 to make it divisible by 8? (a) 2 (b) 6 (c) 5 (d) 7
18. 18.What is the units digit of 3^13? (a) 7 (b) 1 (c) 9 (d) 3
19. 19.Which of these numbers is divisible by 9? (a) 3438 (b) 3445 (c) 3433 (d) 3448
20. 20.What is the remainder when 547 is divided by 10? (a) 6 (b) 7 (c) 3 (d) 8
21. 21.What is the remainder when 801 is divided by 10? (a) 2 (b) 0 (c) 1 (d) 9
22. 22.What is the remainder when 457 is divided by 10? (a) 3 (b) 6 (c) 8 (d) 7
23. 23.What is the smallest number you must add to 251 to make it divisible by 4? (a) 2 (b) 1 (c) 3
24. 24.What is the units digit of 7^14? (a) 3 (b) 7 (c) 9 (d) 1
25. 25.What is the remainder when 130 is divided by 6? (a) 5 (b) 3 (c) 2 (d) 4
26. 26.What is the remainder when 132 is divided by 5? (a) 1 (b) 3 (c) 2 (d) 4
27. 27.What is the units digit of 8^56? (a) 6 (b) 8 (c) 2 (d) 4
28. 28.What is the smallest number you must add to 124 to make it divisible by 3? (a) 1 (b) 2
29. 29.What is the remainder when 778 is divided by 11? (a) 7 (b) 9 (c) 3 (d) 8
30. 30.What is the smallest number you must add to 676 to make it divisible by 10? (a) 5 (b) 6 (c) 3 (d) 4
31. 31.What is the units digit of 2^53? (a) 2 (b) 6 (c) 8 (d) 4
32. 32.Which of these numbers is divisible by 9? (a) 3010 (b) 2997 (c) 2989 (d) 3001
33. 33.What is the smallest number you must add to 343 to make it divisible by 9? (a) 6 (b) 1 (c) 7 (d) 8
34. 34.What is the remainder when 805 is divided by 11? (a) 2 (b) 3 (c) 1 (d) 9
35. 35.What is the remainder when 100 is divided by 3? (a) 2 (b) 1 (c) 0
36. 36.What is the smallest number you must add to 218 to make it divisible by 4? (a) 3 (b) 2 (c) 1
37. 37.What is the smallest number you must add to 304 to make it divisible by 12? (a) 4 (b) 8 (c) 7 (d) 9
38. 38.What is the smallest number you must add to 212 to make it divisible by 3? (a) 2 (b) 1
39. 39.Which of these numbers is divisible by 6? (a) 690 (b) 688 (c) 693 (d) 700
40. 40.What is the units digit of 9^19? (a) 1 (b) 7 (c) 9 (d) 8

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