Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 859 · 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 456 is divided by 9? (a) 3 (b) 7 (c) 6 (d) 5
2. 2.What is the units digit of 2^26? (a) 6 (b) 4 (c) 8 (d) 2
3. 3.Which of these numbers is divisible by 3? (a) 366 (b) 364 (c) 367 (d) 370
4. 4.What is the units digit of 3^32? (a) 9 (b) 1 (c) 3 (d) 7
5. 5.What is the smallest number you must add to 546 to make it divisible by 12? (a) 5 (b) 6 (c) 8 (d) 7
6. 6.Which of these numbers is divisible by 3? (a) 424 (b) 423 (c) 422 (d) 428
7. 7.What is the units digit of 2^13? (a) 6 (b) 4 (c) 8 (d) 2
8. 8.What is the remainder when 321 is divided by 9? (a) 7 (b) 6 (c) 3 (d) 5
9. 9.What is the remainder when 389 is divided by 8? (a) 4 (b) 3 (c) 6 (d) 5
10. 10.Which of these numbers is divisible by 9? (a) 7631 (b) 7607 (c) 7616 (d) 7614
11. 11.What is the smallest number you must add to 159 to make it divisible by 8? (a) 7 (b) 2 (c) 1 (d) 3
12. 12.What is the smallest number you must add to 210 to make it divisible by 4? (a) 1 (b) 2 (c) 3
13. 13.What is the remainder when 888 is divided by 11? (a) 8 (b) 9 (c) 7 (d) 3
14. 14.What is the remainder when 97 is divided by 8? (a) 2 (b) 7 (c) 0 (d) 1
15. 15.What is the units digit of 2^55? (a) 2 (b) 4 (c) 8 (d) 6
16. 16.What is the units digit of 2^58? (a) 2 (b) 8 (c) 6 (d) 4
17. 17.What is the remainder when 269 is divided by 4? (a) 3 (b) 1 (c) 0 (d) 2
18. 18.What is the remainder when 285 is divided by 4? (a) 3 (b) 0 (c) 2 (d) 1
19. 19.What is the units digit of 3^18? (a) 1 (b) 3 (c) 9 (d) 7
20. 20.What is the units digit of 3^14? (a) 1 (b) 3 (c) 7 (d) 9
21. 21.What is the remainder when 712 is divided by 9? (a) 2 (b) 0 (c) 8 (d) 1
22. 22.What is the units digit of 8^58? (a) 6 (b) 4 (c) 2 (d) 8
23. 23.What is the smallest number you must add to 184 to make it divisible by 11? (a) 8 (b) 3 (c) 4 (d) 2
24. 24.Which of these numbers is divisible by 4? (a) 191 (b) 181 (c) 184 (d) 185
25. 25.What is the units digit of 4^24? (a) 7 (b) 4 (c) 6 (d) 5
26. 26.What is the remainder when 267 is divided by 4? (a) 0 (b) 3 (c) 1 (d) 2
27. 27.What is the remainder when 517 is divided by 12? (a) 11 (b) 0 (c) 1 (d) 2
28. 28.Which of these numbers is divisible by 9? (a) 355 (b) 366 (c) 360 (d) 371
29. 29.Which of these numbers is divisible by 11? (a) 5216 (b) 5232 (c) 5246 (d) 5225
30. 30.What is the units digit of 4^27? (a) 4 (b) 3 (c) 5 (d) 6
31. 31.Which of these numbers is divisible by 3? (a) 2162 (b) 2159 (c) 2165 (d) 2160
32. 32.What is the remainder when 214 is divided by 3? (a) 0 (b) 2 (c) 1
33. 33.What is the remainder when 121 is divided by 3? (a) 1 (b) 0 (c) 2
34. 34.What is the remainder when 305 is divided by 4? (a) 0 (b) 1 (c) 2 (d) 3
35. 35.Which of these numbers is divisible by 8? (a) 2728 (b) 2727 (c) 2733 (d) 2740
36. 36.Which of these numbers is divisible by 4? (a) 1147 (b) 1151 (c) 1144 (d) 1143
37. 37.What is the smallest number you must add to 526 to make it divisible by 11? (a) 2 (b) 1 (c) 9 (d) 3
38. 38.What is the remainder when 186 is divided by 5? (a) 1 (b) 4 (c) 2 (d) 0
39. 39.Which of these numbers is divisible by 3? (a) 624 (b) 622 (c) 625 (d) 628
40. 40.What is the smallest number you must add to 291 to make it divisible by 7? (a) 3 (b) 2 (c) 5 (d) 4

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