Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 15 · 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 units digit of 4^19? (a) 5 (b) 3 (c) 6 (d) 4
2. 2.What is the smallest number you must add to 94 to make it divisible by 3? (a) 2 (b) 1
3. 3.What is the remainder when 871 is divided by 11? (a) 9 (b) 2 (c) 1 (d) 3
4. 4.What is the remainder when 301 is divided by 9? (a) 6 (b) 5 (c) 4 (d) 3
5. 5.What is the smallest number you must add to 95 to make it divisible by 3? (a) 1 (b) 2
6. 6.What is the remainder when 699 is divided by 12? (a) 3 (b) 9 (c) 4 (d) 2
7. 7.What is the units digit of 3^20? (a) 3 (b) 9 (c) 1 (d) 7
8. 8.Which of these numbers is divisible by 3? (a) 1387 (b) 1385 (c) 1391 (d) 1386
9. 9.Which of these numbers is divisible by 3? (a) 737 (b) 740 (c) 742 (d) 738
10. 10.What is the units digit of 2^22? (a) 8 (b) 2 (c) 4 (d) 6
11. 11.Which of these numbers is divisible by 4? (a) 624 (b) 622 (c) 627 (d) 629
12. 12.Which of these numbers is divisible by 4? (a) 1929 (b) 1926 (c) 1924 (d) 1923
13. 13.What is the remainder when 85 is divided by 3? (a) 1 (b) 2 (c) 0
14. 14.What is the remainder when 174 is divided by 8? (a) 2 (b) 5 (c) 6 (d) 7
15. 15.Which of these numbers is divisible by 6? (a) 4688 (b) 4686 (c) 4685 (d) 4695
16. 16.What is the remainder when 551 is divided by 10? (a) 9 (b) 0 (c) 2 (d) 1
17. 17.Which of these numbers is divisible by 8? (a) 2137 (b) 2154 (c) 2148 (d) 2144
18. 18.What is the smallest number you must add to 124 to make it divisible by 3? (a) 2 (b) 1
19. 19.Which of these numbers is divisible by 11? (a) 8558 (b) 8574 (c) 8567 (d) 8549
20. 20.Which of these numbers is divisible by 4? (a) 1004 (b) 1006 (c) 1011 (d) 1003
21. 21.What is the remainder when 363 is divided by 9? (a) 2 (b) 4 (c) 3 (d) 6
22. 22.What is the remainder when 356 is divided by 12? (a) 9 (b) 8 (c) 4 (d) 7
23. 23.Which of these numbers is divisible by 8? (a) 1467 (b) 1456 (c) 1463 (d) 1452
24. 24.What is the units digit of 8^45? (a) 6 (b) 2 (c) 8 (d) 4
25. 25.What is the remainder when 151 is divided by 8? (a) 6 (b) 7 (c) 1 (d) 5
26. 26.What is the remainder when 59 is divided by 4? (a) 0 (b) 3 (c) 2 (d) 1
27. 27.What is the smallest number you must add to 256 to make it divisible by 5? (a) 4 (b) 1 (c) 2 (d) 3
28. 28.What is the remainder when 103 is divided by 3? (a) 0 (b) 1 (c) 2
29. 29.Which of these numbers is divisible by 9? (a) 1615 (b) 1606 (c) 1611 (d) 1627
30. 30.What is the smallest number you must add to 673 to make it divisible by 12? (a) 11 (b) 10 (c) 9 (d) 1
31. 31.What is the smallest number you must add to 362 to make it divisible by 9? (a) 2 (b) 8 (c) 6 (d) 7
32. 32.Which of these numbers is divisible by 11? (a) 9534 (b) 9525 (c) 9509 (d) 9515
33. 33.Which of these numbers is divisible by 4? (a) 889 (b) 888 (c) 886 (d) 894
34. 34.What is the remainder when 123 is divided by 7? (a) 4 (b) 6 (c) 5 (d) 3
35. 35.What is the units digit of 4^50? (a) 4 (b) 7 (c) 6 (d) 5
36. 36.What is the units digit of 8^30? (a) 6 (b) 8 (c) 4 (d) 2
37. 37.Which of these numbers is divisible by 3? (a) 758 (b) 763 (c) 759 (d) 760
38. 38.What is the units digit of 9^22? (a) 0 (b) 1 (c) 2 (d) 9
39. 39.What is the smallest number you must add to 281 to make it divisible by 7? (a) 5 (b) 1 (c) 4 (d) 6
40. 40.What is the units digit of 3^15? (a) 9 (b) 3 (c) 7 (d) 1

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