Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 306 · 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 795 is divided by 11? (a) 4 (b) 2 (c) 8 (d) 3
2. 2.What is the remainder when 162 is divided by 5? (a) 4 (b) 1 (c) 2 (d) 3
3. 3.What is the smallest number you must add to 91 to make it divisible by 9? (a) 8 (b) 1 (c) 7 (d) 6
4. 4.Which of these numbers is divisible by 6? (a) 1647 (b) 1644 (c) 1639 (d) 1652
5. 5.Which of these numbers is divisible by 6? (a) 721 (b) 733 (c) 731 (d) 726
6. 6.What is the remainder when 458 is divided by 11? (a) 8 (b) 6 (c) 4 (d) 7
7. 7.What is the remainder when 344 is divided by 5? (a) 1 (b) 4 (c) 3 (d) 2
8. 8.Which of these numbers is divisible by 3? (a) 844 (b) 841 (c) 843 (d) 848
9. 9.Which of these numbers is divisible by 9? (a) 5240 (b) 5224 (c) 5232 (d) 5229
10. 10.What is the smallest number you must add to 631 to make it divisible by 8? (a) 2 (b) 1 (c) 7 (d) 3
11. 11.Which of these numbers is divisible by 3? (a) 1079 (b) 1077 (c) 1082 (d) 1076
12. 12.What is the smallest number you must add to 166 to make it divisible by 7? (a) 3 (b) 1 (c) 5 (d) 2
13. 13.What is the remainder when 163 is divided by 7? (a) 5 (b) 3 (c) 1 (d) 2
14. 14.What is the smallest number you must add to 437 to make it divisible by 10? (a) 4 (b) 3 (c) 7 (d) 2
15. 15.What is the units digit of 4^31? (a) 6 (b) 5 (c) 4 (d) 3
16. 16.Which of these numbers is divisible by 4? (a) 1943 (b) 1934 (c) 1937 (d) 1936
17. 17.Which of these numbers is divisible by 4? (a) 1453 (b) 1463 (c) 1456 (d) 1459
18. 18.Which of these numbers is divisible by 8? (a) 1820 (b) 1835 (c) 1829 (d) 1824
19. 19.What is the units digit of 8^14? (a) 8 (b) 4 (c) 6 (d) 2
20. 20.What is the units digit of 9^20? (a) 0 (b) 9 (c) 2 (d) 1
21. 21.What is the units digit of 8^17? (a) 2 (b) 8 (c) 4 (d) 6
22. 22.What is the units digit of 7^41? (a) 7 (b) 3 (c) 1 (d) 9
23. 23.Which of these numbers is divisible by 8? (a) 2612 (b) 2625 (c) 2620 (d) 2616
24. 24.What is the remainder when 92 is divided by 3? (a) 1 (b) 0 (c) 2
25. 25.What is the smallest number you must add to 585 to make it divisible by 12? (a) 3 (b) 2 (c) 4 (d) 9
26. 26.Which of these numbers is divisible by 9? (a) 5022 (b) 5018 (c) 5027 (d) 5036
27. 27.What is the remainder when 729 is divided by 12? (a) 8 (b) 9 (c) 10 (d) 3
28. 28.What is the smallest number you must add to 603 to make it divisible by 11? (a) 3 (b) 9 (c) 1 (d) 2
29. 29.What is the units digit of 4^57? (a) 3 (b) 6 (c) 5 (d) 4
30. 30.What is the units digit of 2^56? (a) 4 (b) 6 (c) 2 (d) 8
31. 31.Which of these numbers is divisible by 6? (a) 4154 (b) 4149 (c) 4146 (d) 4144
32. 32.Which of these numbers is divisible by 3? (a) 269 (b) 265 (c) 267 (d) 271
33. 33.What is the smallest number you must add to 354 to make it divisible by 12? (a) 7 (b) 8 (c) 5 (d) 6
34. 34.What is the smallest number you must add to 32 to make it divisible by 3? (a) 2 (b) 1
35. 35.What is the smallest number you must add to 205 to make it divisible by 6? (a) 3 (b) 5 (c) 4 (d) 1
36. 36.What is the units digit of 4^35? (a) 6 (b) 5 (c) 4 (d) 3
37. 37.What is the units digit of 8^35? (a) 2 (b) 6 (c) 8 (d) 4
38. 38.Which of these numbers is divisible by 9? (a) 7621 (b) 7623 (c) 7626 (d) 7635
39. 39.Which of these numbers is divisible by 4? (a) 3454 (b) 3456 (c) 3457 (d) 3462
40. 40.What is the units digit of 9^16? (a) 0 (b) 9 (c) 2 (d) 1

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