Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 884 · 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 9? (a) 634 (b) 642 (c) 630 (d) 629
2. 2.Which of these numbers is divisible by 11? (a) 3992 (b) 3993 (c) 4005 (d) 3996
3. 3.Which of these numbers is divisible by 4? (a) 1734 (b) 1729 (c) 1728 (d) 1726
4. 4.Which of these numbers is divisible by 4? (a) 2560 (b) 2566 (c) 2558 (d) 2563
5. 5.What is the smallest number you must add to 207 to make it divisible by 6? (a) 3 (b) 5 (c) 2 (d) 4
6. 6.What is the remainder when 167 is divided by 5? (a) 2 (b) 1 (c) 4 (d) 3
7. 7.What is the smallest number you must add to 366 to make it divisible by 7? (a) 6 (b) 2 (c) 5 (d) 4
8. 8.Which of these numbers is divisible by 8? (a) 7193 (b) 7204 (c) 7212 (d) 7200
9. 9.What is the remainder when 205 is divided by 9? (a) 8 (b) 6 (c) 7 (d) 2
10. 10.What is the remainder when 175 is divided by 8? (a) 1 (b) 7 (c) 5 (d) 6
11. 11.What is the remainder when 502 is divided by 11? (a) 4 (b) 6 (c) 7 (d) 8
12. 12.What is the remainder when 163 is divided by 8? (a) 2 (b) 4 (c) 3 (d) 5
13. 13.What is the units digit of 3^42? (a) 3 (b) 7 (c) 9 (d) 1
14. 14.What is the smallest number you must add to 491 to make it divisible by 8? (a) 4 (b) 6 (c) 5 (d) 3
15. 15.What is the smallest number you must add to 191 to make it divisible by 7? (a) 2 (b) 4 (c) 5 (d) 6
16. 16.What is the remainder when 175 is divided by 10? (a) 7 (b) 5 (c) 4 (d) 6
17. 17.Which of these numbers is divisible by 3? (a) 1924 (b) 1918 (c) 1922 (d) 1920
18. 18.Which of these numbers is divisible by 3? (a) 1660 (b) 1657 (c) 1659 (d) 1663
19. 19.What is the units digit of 4^12? (a) 4 (b) 5 (c) 7 (d) 6
20. 20.Which of these numbers is divisible by 9? (a) 6654 (b) 6648 (c) 6651 (d) 6666
21. 21.Which of these numbers is divisible by 3? (a) 1355 (b) 1357 (c) 1356 (d) 1360
22. 22.Which of these numbers is divisible by 9? (a) 1672 (b) 1685 (c) 1674 (d) 1676
23. 23.Which of these numbers is divisible by 11? (a) 5805 (b) 5783 (c) 5787 (d) 5786
24. 24.Which of these numbers is divisible by 6? (a) 4235 (b) 4243 (c) 4240 (d) 4236
25. 25.What is the smallest number you must add to 241 to make it divisible by 3? (a) 1 (b) 2
26. 26.Which of these numbers is divisible by 4? (a) 971 (b) 975 (c) 972 (d) 977
27. 27.What is the smallest number you must add to 271 to make it divisible by 10? (a) 1 (b) 9 (c) 7 (d) 8
28. 28.What is the remainder when 122 is divided by 12? (a) 1 (b) 3 (c) 2 (d) 10
29. 29.What is the units digit of 9^41? (a) 1 (b) 8 (c) 9 (d) 7
30. 30.What is the smallest number you must add to 52 to make it divisible by 5? (a) 1 (b) 2 (c) 3 (d) 4
31. 31.Which of these numbers is divisible by 4? (a) 1416 (b) 1414 (c) 1423 (d) 1417
32. 32.What is the remainder when 695 is divided by 10? (a) 5 (b) 7 (c) 4 (d) 6
33. 33.What is the units digit of 3^52? (a) 1 (b) 7 (c) 9 (d) 3
34. 34.Which of these numbers is divisible by 6? (a) 407 (b) 408 (c) 409 (d) 416
35. 35.What is the remainder when 502 is divided by 12? (a) 10 (b) 11 (c) 9 (d) 2
36. 36.Which of these numbers is divisible by 3? (a) 651 (b) 653 (c) 649 (d) 656
37. 37.What is the remainder when 82 is divided by 3? (a) 0 (b) 1 (c) 2
38. 38.What is the remainder when 162 is divided by 4? (a) 0 (b) 2 (c) 1 (d) 3
39. 39.What is the units digit of 4^34? (a) 7 (b) 4 (c) 5 (d) 6
40. 40.What is the remainder when 626 is divided by 10? (a) 4 (b) 7 (c) 5 (d) 6

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