Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 671 · 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.Which of these numbers is divisible by 4? (a) 530 (b) 532 (c) 534 (d) 537
2. 2.What is the remainder when 131 is divided by 3? (a) 1 (b) 0 (c) 2
3. 3.Which of these numbers is divisible by 11? (a) 4727 (b) 4708 (c) 4706 (d) 4711
4. 4.Which of these numbers is divisible by 11? (a) 1062 (b) 1036 (c) 1045 (d) 1049
5. 5.Which of these numbers is divisible by 11? (a) 8354 (b) 8349 (c) 8348 (d) 8365
6. 6.What is the smallest number you must add to 41 to make it divisible by 4? (a) 2 (b) 1 (c) 3
7. 7.What is the remainder when 534 is divided by 7? (a) 5 (b) 2 (c) 3 (d) 1
8. 8.What is the units digit of 8^31? (a) 4 (b) 2 (c) 8 (d) 6
9. 9.Which of these numbers is divisible by 8? (a) 1623 (b) 1608 (c) 1612 (d) 1607
10. 10.What is the smallest number you must add to 328 to make it divisible by 5? (a) 3 (b) 4 (c) 2 (d) 1
11. 11.What is the remainder when 178 is divided by 9? (a) 2 (b) 8 (c) 6 (d) 7
12. 12.What is the remainder when 225 is divided by 8? (a) 7 (b) 1 (c) 0 (d) 2
13. 13.What is the remainder when 198 is divided by 4? (a) 0 (b) 2 (c) 1 (d) 3
14. 14.What is the units digit of 9^35? (a) 7 (b) 9 (c) 8 (d) 1
15. 15.Which of these numbers is divisible by 8? (a) 4376 (b) 4375 (c) 4386 (d) 4378
16. 16.What is the units digit of 3^31? (a) 3 (b) 9 (c) 7 (d) 1
17. 17.What is the remainder when 374 is divided by 8? (a) 7 (b) 2 (c) 6 (d) 5
18. 18.What is the units digit of 8^15? (a) 2 (b) 6 (c) 4 (d) 8
19. 19.What is the smallest number you must add to 374 to make it divisible by 6? (a) 2 (b) 5 (c) 3 (d) 4
20. 20.What is the smallest number you must add to 271 to make it divisible by 8? (a) 2 (b) 3 (c) 1 (d) 7
21. 21.Which of these numbers is divisible by 9? (a) 4176 (b) 4180 (c) 4192 (d) 4171
22. 22.Which of these numbers is divisible by 9? (a) 3663 (b) 3678 (c) 3655 (d) 3666
23. 23.What is the units digit of 4^18? (a) 7 (b) 5 (c) 4 (d) 6
24. 24.Which of these numbers is divisible by 9? (a) 1229 (b) 1237 (c) 1224 (d) 1216
25. 25.What is the units digit of 8^39? (a) 2 (b) 8 (c) 6 (d) 4
26. 26.What is the remainder when 221 is divided by 9? (a) 5 (b) 7 (c) 4 (d) 6
27. 27.What is the remainder when 70 is divided by 4? (a) 2 (b) 3 (c) 1 (d) 0
28. 28.Which of these numbers is divisible by 3? (a) 1635 (b) 1633 (c) 1636 (d) 1640
29. 29.What is the remainder when 595 is divided by 12? (a) 8 (b) 6 (c) 5 (d) 7
30. 30.What is the remainder when 855 is divided by 12? (a) 9 (b) 2 (c) 3 (d) 4
31. 31.Which of these numbers is divisible by 11? (a) 1210 (b) 1231 (c) 1201 (d) 1215
32. 32.What is the units digit of 3^29? (a) 1 (b) 7 (c) 9 (d) 3
33. 33.What is the smallest number you must add to 316 to make it divisible by 5? (a) 4 (b) 1 (c) 3 (d) 2
34. 34.What is the remainder when 149 is divided by 12? (a) 7 (b) 4 (c) 5 (d) 6
35. 35.What is the smallest number you must add to 194 to make it divisible by 12? (a) 9 (b) 10 (c) 11 (d) 2
36. 36.What is the remainder when 743 is divided by 11? (a) 6 (b) 7 (c) 8 (d) 5
37. 37.What is the remainder when 777 is divided by 12? (a) 3 (b) 10 (c) 9 (d) 8
38. 38.What is the smallest number you must add to 474 to make it divisible by 9? (a) 4 (b) 2 (c) 6 (d) 3
39. 39.What is the smallest number you must add to 129 to make it divisible by 9? (a) 5 (b) 3 (c) 7 (d) 6
40. 40.What is the remainder when 391 is divided by 5? (a) 2 (b) 1 (c) 0 (d) 4

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