Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 322 · 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 699 is divided by 9? (a) 3 (b) 6 (c) 5 (d) 7
2. 2.What is the remainder when 444 is divided by 10? (a) 3 (b) 5 (c) 6 (d) 4
3. 3.Which of these numbers is divisible by 9? (a) 6199 (b) 6206 (c) 6213 (d) 6201
4. 4.What is the remainder when 258 is divided by 4? (a) 3 (b) 1 (c) 0 (d) 2
5. 5.Which of these numbers is divisible by 9? (a) 3341 (b) 3359 (c) 3348 (d) 3353
6. 6.What is the units digit of 7^25? (a) 9 (b) 1 (c) 3 (d) 7
7. 7.What is the remainder when 88 is divided by 5? (a) 3 (b) 1 (c) 2 (d) 4
8. 8.What is the smallest number you must add to 258 to make it divisible by 10? (a) 3 (b) 8 (c) 2 (d) 1
9. 9.What is the smallest number you must add to 219 to make it divisible by 6? (a) 3 (b) 5 (c) 2 (d) 4
10. 10.What is the remainder when 587 is divided by 11? (a) 3 (b) 7 (c) 4 (d) 5
11. 11.What is the units digit of 7^53? (a) 1 (b) 3 (c) 9 (d) 7
12. 12.Which of these numbers is divisible by 11? (a) 1692 (b) 1694 (c) 1704 (d) 1711
13. 13.What is the smallest number you must add to 232 to make it divisible by 12? (a) 7 (b) 8 (c) 9 (d) 4
14. 14.What is the remainder when 235 is divided by 6? (a) 1 (b) 0 (c) 5 (d) 2
15. 15.What is the smallest number you must add to 238 to make it divisible by 3? (a) 1 (b) 2
16. 16.Which of these numbers is divisible by 3? (a) 286 (b) 283 (c) 285 (d) 290
17. 17.Which of these numbers is divisible by 11? (a) 7470 (b) 7458 (c) 7450 (d) 7462
18. 18.What is the remainder when 170 is divided by 7? (a) 2 (b) 5 (c) 1 (d) 3
19. 19.What is the smallest number you must add to 148 to make it divisible by 10? (a) 8 (b) 3 (c) 1 (d) 2
20. 20.What is the remainder when 255 is divided by 6? (a) 2 (b) 4 (c) 3 (d) 5
21. 21.What is the units digit of 7^16? (a) 1 (b) 7 (c) 3 (d) 9
22. 22.What is the units digit of 9^23? (a) 7 (b) 9 (c) 8 (d) 1
23. 23.Which of these numbers is divisible by 3? (a) 1050 (b) 1051 (c) 1055 (d) 1049
24. 24.Which of these numbers is divisible by 6? (a) 3062 (b) 3076 (c) 3068 (d) 3066
25. 25.What is the smallest number you must add to 599 to make it divisible by 8? (a) 7 (b) 1 (c) 3 (d) 2
26. 26.What is the smallest number you must add to 227 to make it divisible by 5? (a) 2 (b) 3 (c) 1 (d) 4
27. 27.Which of these numbers is divisible by 3? (a) 1178 (b) 1181 (c) 1174 (d) 1176
28. 28.Which of these numbers is divisible by 11? (a) 6908 (b) 6906 (c) 6925 (d) 6912
29. 29.What is the smallest number you must add to 451 to make it divisible by 10? (a) 7 (b) 8 (c) 1 (d) 9
30. 30.Which of these numbers is divisible by 9? (a) 4269 (b) 4257 (c) 4265 (d) 4252
31. 31.What is the smallest number you must add to 268 to make it divisible by 7? (a) 6 (b) 4 (c) 2 (d) 5
32. 32.What is the remainder when 449 is divided by 6? (a) 4 (b) 1 (c) 5 (d) 3
33. 33.What is the units digit of 3^56? (a) 3 (b) 1 (c) 7 (d) 9
34. 34.Which of these numbers is divisible by 6? (a) 635 (b) 619 (c) 628 (d) 624
35. 35.What is the remainder when 405 is divided by 12? (a) 9 (b) 8 (c) 3 (d) 10
36. 36.What is the smallest number you must add to 314 to make it divisible by 6? (a) 2 (b) 4 (c) 5 (d) 3
37. 37.What is the remainder when 395 is divided by 8? (a) 5 (b) 4 (c) 2 (d) 3
38. 38.What is the units digit of 9^15? (a) 8 (b) 1 (c) 7 (d) 9
39. 39.What is the units digit of 9^16? (a) 0 (b) 2 (c) 1 (d) 9
40. 40.What is the remainder when 522 is divided by 11? (a) 4 (b) 6 (c) 7 (d) 5

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