Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 501 · 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 295 to make it divisible by 7? (a) 4 (b) 6 (c) 5 (d) 1
2. 2.What is the smallest number you must add to 176 to make it divisible by 6? (a) 4 (b) 2 (c) 3 (d) 5
3. 3.What is the units digit of 3^55? (a) 1 (b) 3 (c) 9 (d) 7
4. 4.Which of these numbers is divisible by 6? (a) 5149 (b) 5142 (c) 5141 (d) 5146
5. 5.What is the remainder when 181 is divided by 5? (a) 4 (b) 2 (c) 0 (d) 1
6. 6.Which of these numbers is divisible by 11? (a) 6369 (b) 6366 (c) 6371 (d) 6383
7. 7.What is the remainder when 229 is divided by 10? (a) 8 (b) 1 (c) 9 (d) 7
8. 8.Which of these numbers is divisible by 3? (a) 2631 (b) 2629 (c) 2632 (d) 2635
9. 9.Which of these numbers is divisible by 11? (a) 6274 (b) 6270 (c) 6269 (d) 6291
10. 10.What is the remainder when 146 is divided by 3? (a) 0 (b) 1 (c) 2
11. 11.Which of these numbers is divisible by 9? (a) 4403 (b) 4392 (c) 4391 (d) 4395
12. 12.What is the remainder when 432 is divided by 10? (a) 2 (b) 8 (c) 1 (d) 3
13. 13.What is the remainder when 789 is divided by 11? (a) 7 (b) 9 (c) 8 (d) 3
14. 14.Which of these numbers is divisible by 3? (a) 1444 (b) 1438 (c) 1441 (d) 1440
15. 15.What is the remainder when 198 is divided by 7? (a) 3 (b) 1 (c) 5 (d) 2
16. 16.What is the remainder when 830 is divided by 11? (a) 7 (b) 5 (c) 6 (d) 4
17. 17.What is the remainder when 269 is divided by 5? (a) 3 (b) 1 (c) 4 (d) 2
18. 18.What is the smallest number you must add to 53 to make it divisible by 4? (a) 1 (b) 2 (c) 3
19. 19.What is the smallest number you must add to 264 to make it divisible by 5? (a) 4 (b) 3 (c) 1 (d) 2
20. 20.Which of these numbers is divisible by 4? (a) 288 (b) 285 (c) 293 (d) 289
21. 21.Which of these numbers is divisible by 4? (a) 1046 (b) 1050 (c) 1053 (d) 1048
22. 22.What is the units digit of 4^53? (a) 4 (b) 5 (c) 6 (d) 3
23. 23.What is the remainder when 186 is divided by 5? (a) 1 (b) 4 (c) 2 (d) 0
24. 24.What is the remainder when 218 is divided by 8? (a) 6 (b) 3 (c) 2 (d) 1
25. 25.Which of these numbers is divisible by 11? (a) 814 (b) 828 (c) 815 (d) 807
26. 26.What is the units digit of 3^38? (a) 9 (b) 3 (c) 7 (d) 1
27. 27.What is the remainder when 96 is divided by 7? (a) 6 (b) 2 (c) 4 (d) 5
28. 28.What is the smallest number you must add to 593 to make it divisible by 9? (a) 8 (b) 3 (c) 2 (d) 1
29. 29.Which of these numbers is divisible by 8? (a) 6536 (b) 6529 (c) 6551 (d) 6538
30. 30.What is the smallest number you must add to 292 to make it divisible by 11? (a) 6 (b) 4 (c) 5 (d) 7
31. 31.What is the smallest number you must add to 107 to make it divisible by 3? (a) 2 (b) 1
32. 32.Which of these numbers is divisible by 6? (a) 390 (b) 398 (c) 388 (d) 392
33. 33.What is the units digit of 2^52? (a) 8 (b) 2 (c) 4 (d) 6
34. 34.What is the remainder when 334 is divided by 6? (a) 5 (b) 2 (c) 4 (d) 3
35. 35.What is the smallest number you must add to 202 to make it divisible by 3? (a) 1 (b) 2
36. 36.What is the remainder when 239 is divided by 10? (a) 8 (b) 1 (c) 9 (d) 7
37. 37.What is the remainder when 328 is divided by 11? (a) 9 (b) 8 (c) 10 (d) 2
38. 38.What is the smallest number you must add to 88 to make it divisible by 3? (a) 1 (b) 2
39. 39.What is the smallest number you must add to 699 to make it divisible by 12? (a) 10 (b) 3 (c) 9 (d) 8
40. 40.What is the remainder when 127 is divided by 4? (a) 1 (b) 0 (c) 2 (d) 3

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