Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 758 · 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 902 to make it divisible by 12? (a) 9 (b) 10 (c) 11 (d) 2
2. 2.What is the units digit of 8^56? (a) 2 (b) 6 (c) 4 (d) 8
3. 3.Which of these numbers is divisible by 9? (a) 4541 (b) 4545 (c) 4552 (d) 4561
4. 4.Which of these numbers is divisible by 11? (a) 1573 (b) 1569 (c) 1586 (d) 1575
5. 5.Which of these numbers is divisible by 4? (a) 1645 (b) 1655 (c) 1649 (d) 1648
6. 6.What is the remainder when 236 is divided by 3? (a) 0 (b) 1 (c) 2
7. 7.Which of these numbers is divisible by 4? (a) 2203 (b) 2207 (c) 2204 (d) 2209
8. 8.What is the smallest number you must add to 371 to make it divisible by 5? (a) 2 (b) 3 (c) 4 (d) 1
9. 9.Which of these numbers is divisible by 3? (a) 536 (b) 532 (c) 534 (d) 539
10. 10.What is the units digit of 7^52? (a) 3 (b) 7 (c) 1 (d) 9
11. 11.What is the remainder when 391 is divided by 12? (a) 8 (b) 5 (c) 7 (d) 6
12. 12.What is the remainder when 563 is divided by 12? (a) 1 (b) 10 (c) 9 (d) 11
13. 13.What is the smallest number you must add to 235 to make it divisible by 7? (a) 5 (b) 2 (c) 4 (d) 3
14. 14.What is the remainder when 239 is divided by 9? (a) 6 (b) 5 (c) 7 (d) 4
15. 15.What is the remainder when 125 is divided by 3? (a) 1 (b) 2 (c) 0
16. 16.What is the remainder when 465 is divided by 11? (a) 4 (b) 3 (c) 8 (d) 2
17. 17.Which of these numbers is divisible by 3? (a) 1798 (b) 1796 (c) 1797 (d) 1801
18. 18.Which of these numbers is divisible by 9? (a) 4749 (b) 4736 (c) 4732 (d) 4734
19. 19.What is the remainder when 75 is divided by 6? (a) 4 (b) 2 (c) 3 (d) 5
20. 20.What is the remainder when 817 is divided by 12? (a) 11 (b) 1 (c) 2 (d) 0
21. 21.What is the smallest number you must add to 399 to make it divisible by 5? (a) 3 (b) 1 (c) 2 (d) 4
22. 22.What is the remainder when 328 is divided by 11? (a) 2 (b) 10 (c) 9 (d) 8
23. 23.What is the smallest number you must add to 766 to make it divisible by 10? (a) 4 (b) 3 (c) 5 (d) 6
24. 24.What is the units digit of 3^16? (a) 9 (b) 1 (c) 3 (d) 7
25. 25.Which of these numbers is divisible by 3? (a) 1016 (b) 1012 (c) 1019 (d) 1014
26. 26.Which of these numbers is divisible by 8? (a) 1757 (b) 1765 (c) 1760 (d) 1770
27. 27.What is the remainder when 329 is divided by 12? (a) 7 (b) 5 (c) 4 (d) 6
28. 28.Which of these numbers is divisible by 11? (a) 4439 (b) 4451 (c) 4461 (d) 4444
29. 29.What is the units digit of 3^35? (a) 9 (b) 7 (c) 1 (d) 3
30. 30.What is the units digit of 3^30? (a) 1 (b) 7 (c) 9 (d) 3
31. 31.What is the smallest number you must add to 76 to make it divisible by 5? (a) 4 (b) 2 (c) 3 (d) 1
32. 32.Which of these numbers is divisible by 11? (a) 6807 (b) 6821 (c) 6816 (d) 6809
33. 33.Which of these numbers is divisible by 8? (a) 7038 (b) 7046 (c) 7040 (d) 7050
34. 34.What is the remainder when 427 is divided by 9? (a) 4 (b) 5 (c) 6 (d) 3
35. 35.Which of these numbers is divisible by 11? (a) 2824 (b) 2837 (c) 2816 (d) 2807
36. 36.What is the units digit of 9^32? (a) 9 (b) 2 (c) 0 (d) 1
37. 37.What is the remainder when 229 is divided by 3? (a) 2 (b) 0 (c) 1
38. 38.What is the remainder when 70 is divided by 4? (a) 0 (b) 1 (c) 3 (d) 2
39. 39.What is the units digit of 4^52? (a) 6 (b) 5 (c) 4 (d) 7
40. 40.What is the smallest number you must add to 68 to make it divisible by 5? (a) 4 (b) 3 (c) 1 (d) 2

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