Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 291 · 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 284 to make it divisible by 12? (a) 8 (b) 3 (c) 5 (d) 4
2. 2.What is the remainder when 922 is divided by 12? (a) 9 (b) 2 (c) 10 (d) 11
3. 3.Which of these numbers is divisible by 11? (a) 7788 (b) 7805 (c) 7790 (d) 7786
4. 4.What is the remainder when 484 is divided by 8? (a) 5 (b) 3 (c) 4 (d) 6
5. 5.What is the smallest number you must add to 411 to make it divisible by 7? (a) 3 (b) 2 (c) 5 (d) 1
6. 6.What is the units digit of 3^30? (a) 3 (b) 7 (c) 1 (d) 9
7. 7.What is the units digit of 8^37? (a) 4 (b) 2 (c) 8 (d) 6
8. 8.What is the remainder when 247 is divided by 8? (a) 6 (b) 7 (c) 1 (d) 5
9. 9.Which of these numbers is divisible by 4? (a) 307 (b) 302 (c) 304 (d) 311
10. 10.What is the smallest number you must add to 213 to make it divisible by 5? (a) 1 (b) 3 (c) 4 (d) 2
11. 11.What is the units digit of 2^24? (a) 4 (b) 2 (c) 8 (d) 6
12. 12.What is the remainder when 134 is divided by 4? (a) 1 (b) 3 (c) 0 (d) 2
13. 13.Which of these numbers is divisible by 11? (a) 3214 (b) 3230 (c) 3239 (d) 3223
14. 14.Which of these numbers is divisible by 11? (a) 5566 (b) 5568 (c) 5581 (d) 5563
15. 15.What is the smallest number you must add to 249 to make it divisible by 5? (a) 4 (b) 2 (c) 3 (d) 1
16. 16.What is the remainder when 145 is divided by 4? (a) 3 (b) 0 (c) 1 (d) 2
17. 17.What is the units digit of 3^25? (a) 3 (b) 7 (c) 9 (d) 1
18. 18.What is the units digit of 8^56? (a) 6 (b) 8 (c) 2 (d) 4
19. 19.What is the smallest number you must add to 195 to make it divisible by 7? (a) 3 (b) 1 (c) 6 (d) 2
20. 20.What is the remainder when 505 is divided by 12? (a) 2 (b) 1 (c) 11 (d) 0
21. 21.What is the units digit of 3^55? (a) 7 (b) 1 (c) 3 (d) 9
22. 22.What is the smallest number you must add to 486 to make it divisible by 12? (a) 7 (b) 6 (c) 8 (d) 5
23. 23.What is the remainder when 293 is divided by 4? (a) 1 (b) 0 (c) 2 (d) 3
24. 24.What is the remainder when 401 is divided by 7? (a) 1 (b) 5 (c) 3 (d) 2
25. 25.What is the smallest number you must add to 369 to make it divisible by 11? (a) 4 (b) 6 (c) 7 (d) 5
26. 26.What is the remainder when 35 is divided by 3? (a) 1 (b) 0 (c) 2
27. 27.What is the remainder when 485 is divided by 7? (a) 2 (b) 1 (c) 3 (d) 5
28. 28.Which of these numbers is divisible by 3? (a) 374 (b) 377 (c) 380 (d) 375
29. 29.What is the smallest number you must add to 334 to make it divisible by 8? (a) 6 (b) 3 (c) 1 (d) 2
30. 30.What is the smallest number you must add to 137 to make it divisible by 10? (a) 7 (b) 3 (c) 4 (d) 2
31. 31.What is the remainder when 658 is divided by 9? (a) 1 (b) 2 (c) 0 (d) 8
32. 32.What is the smallest number you must add to 165 to make it divisible by 6? (a) 2 (b) 3 (c) 4 (d) 5
33. 33.Which of these numbers is divisible by 6? (a) 394 (b) 386 (c) 400 (d) 390
34. 34.Which of these numbers is divisible by 3? (a) 917 (b) 915 (c) 913 (d) 920
35. 35.What is the units digit of 8^59? (a) 4 (b) 6 (c) 2 (d) 8
36. 36.What is the smallest number you must add to 337 to make it divisible by 6? (a) 5 (b) 3 (c) 4 (d) 1
37. 37.What is the remainder when 359 is divided by 7? (a) 5 (b) 2 (c) 3 (d) 1
38. 38.What is the remainder when 89 is divided by 4? (a) 3 (b) 2 (c) 1 (d) 0
39. 39.What is the remainder when 848 is divided by 11? (a) 1 (b) 10 (c) 2 (d) 0
40. 40.What is the remainder when 201 is divided by 6? (a) 4 (b) 5 (c) 3 (d) 2

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