Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 595 · 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 2^14? (a) 2 (b) 8 (c) 6 (d) 4
2. 2.What is the remainder when 732 is divided by 10? (a) 1 (b) 3 (c) 2 (d) 8
3. 3.What is the units digit of 8^36? (a) 2 (b) 4 (c) 8 (d) 6
4. 4.What is the remainder when 503 is divided by 7? (a) 6 (b) 1 (c) 5 (d) 4
5. 5.What is the remainder when 247 is divided by 8? (a) 7 (b) 1 (c) 5 (d) 6
6. 6.Which of these numbers is divisible by 8? (a) 5446 (b) 5430 (c) 5432 (d) 5436
7. 7.What is the remainder when 615 is divided by 10? (a) 7 (b) 6 (c) 5 (d) 4
8. 8.What is the smallest number you must add to 427 to make it divisible by 9? (a) 6 (b) 7 (c) 4 (d) 5
9. 9.What is the smallest number you must add to 193 to make it divisible by 9? (a) 4 (b) 5 (c) 7 (d) 6
10. 10.What is the smallest number you must add to 202 to make it divisible by 7? (a) 6 (b) 2 (c) 1 (d) 3
11. 11.Which of these numbers is divisible by 9? (a) 2864 (b) 2877 (c) 2888 (d) 2871
12. 12.What is the smallest number you must add to 305 to make it divisible by 11? (a) 8 (b) 2 (c) 3 (d) 4
13. 13.What is the remainder when 223 is divided by 3? (a) 1 (b) 2 (c) 0
14. 14.Which of these numbers is divisible by 4? (a) 2549 (b) 2545 (c) 2541 (d) 2544
15. 15.What is the remainder when 722 is divided by 10? (a) 1 (b) 2 (c) 3 (d) 8
16. 16.What is the remainder when 281 is divided by 5? (a) 1 (b) 4 (c) 2 (d) 0
17. 17.Which of these numbers is divisible by 4? (a) 1256 (b) 1258 (c) 1262 (d) 1254
18. 18.What is the units digit of 3^21? (a) 3 (b) 1 (c) 7 (d) 9
19. 19.What is the remainder when 348 is divided by 5? (a) 3 (b) 1 (c) 4 (d) 2
20. 20.What is the smallest number you must add to 809 to make it divisible by 11? (a) 6 (b) 7 (c) 4 (d) 5
21. 21.Which of these numbers is divisible by 8? (a) 5196 (b) 5184 (c) 5190 (d) 5181
22. 22.Which of these numbers is divisible by 6? (a) 3921 (b) 3914 (c) 3918 (d) 3928
23. 23.What is the remainder when 104 is divided by 9? (a) 7 (b) 4 (c) 5 (d) 6
24. 24.Which of these numbers is divisible by 8? (a) 6973 (b) 6968 (c) 6966 (d) 6981
25. 25.What is the units digit of 2^30? (a) 4 (b) 8 (c) 6 (d) 2
26. 26.What is the remainder when 235 is divided by 8? (a) 2 (b) 5 (c) 4 (d) 3
27. 27.What is the remainder when 63 is divided by 4? (a) 2 (b) 1 (c) 0 (d) 3
28. 28.Which of these numbers is divisible by 11? (a) 8119 (b) 8107 (c) 8098 (d) 8109
29. 29.What is the remainder when 302 is divided by 12? (a) 3 (b) 1 (c) 10 (d) 2
30. 30.Which of these numbers is divisible by 9? (a) 405 (b) 421 (c) 413 (d) 400
31. 31.What is the units digit of 7^14? (a) 3 (b) 7 (c) 9 (d) 1
32. 32.What is the smallest number you must add to 277 to make it divisible by 9? (a) 7 (b) 1 (c) 2 (d) 3
33. 33.Which of these numbers is divisible by 4? (a) 211 (b) 204 (c) 203 (d) 206
34. 34.What is the remainder when 95 is divided by 3? (a) 2 (b) 0 (c) 1
35. 35.Which of these numbers is divisible by 11? (a) 4251 (b) 4232 (c) 4244 (d) 4235
36. 36.What is the units digit of 7^24? (a) 7 (b) 3 (c) 1 (d) 9
37. 37.Which of these numbers is divisible by 3? (a) 562 (b) 565 (c) 564 (d) 569
38. 38.What is the remainder when 276 is divided by 5? (a) 4 (b) 2 (c) 1 (d) 0
39. 39.Which of these numbers is divisible by 4? (a) 1912 (b) 1917 (c) 1914 (d) 1909
40. 40.Which of these numbers is divisible by 11? (a) 9509 (b) 9515 (c) 9520 (d) 9529

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