Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 640 · 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 830 to make it divisible by 12? (a) 2 (b) 11 (c) 10 (d) 9
2. 2.What is the smallest number you must add to 244 to make it divisible by 5? (a) 2 (b) 1 (c) 4 (d) 3
3. 3.What is the remainder when 111 is divided by 5? (a) 2 (b) 4 (c) 0 (d) 1
4. 4.What is the remainder when 369 is divided by 5? (a) 2 (b) 1 (c) 4 (d) 3
5. 5.What is the remainder when 876 is divided by 11? (a) 7 (b) 4 (c) 8 (d) 6
6. 6.What is the smallest number you must add to 313 to make it divisible by 4? (a) 2 (b) 1 (c) 3
7. 7.What is the remainder when 227 is divided by 4? (a) 1 (b) 2 (c) 0 (d) 3
8. 8.What is the units digit of 9^38? (a) 1 (b) 2 (c) 0 (d) 9
9. 9.What is the units digit of 4^24? (a) 4 (b) 7 (c) 5 (d) 6
10. 10.What is the remainder when 188 is divided by 7? (a) 6 (b) 1 (c) 4 (d) 5
11. 11.What is the smallest number you must add to 85 to make it divisible by 8? (a) 2 (b) 4 (c) 3 (d) 5
12. 12.Which of these numbers is divisible by 3? (a) 1963 (b) 1967 (c) 1962 (d) 1960
13. 13.What is the smallest number you must add to 192 to make it divisible by 10? (a) 7 (b) 9 (c) 8 (d) 2
14. 14.Which of these numbers is divisible by 8? (a) 5368 (b) 5380 (c) 5366 (d) 5374
15. 15.Which of these numbers is divisible by 11? (a) 9898 (b) 9876 (c) 9878 (d) 9881
16. 16.What is the smallest number you must add to 114 to make it divisible by 11? (a) 6 (b) 4 (c) 8 (d) 7
17. 17.What is the units digit of 7^28? (a) 9 (b) 3 (c) 7 (d) 1
18. 18.What is the smallest number you must add to 218 to make it divisible by 3? (a) 2 (b) 1
19. 19.Which of these numbers is divisible by 3? (a) 1774 (b) 1770 (c) 1771 (d) 1768
20. 20.What is the units digit of 3^27? (a) 1 (b) 3 (c) 9 (d) 7
21. 21.What is the units digit of 9^41? (a) 9 (b) 8 (c) 1 (d) 7
22. 22.Which of these numbers is divisible by 11? (a) 5940 (b) 5959 (c) 5931 (d) 5942
23. 23.What is the smallest number you must add to 182 to make it divisible by 3? (a) 1 (b) 2
24. 24.What is the smallest number you must add to 513 to make it divisible by 10? (a) 6 (b) 7 (c) 3 (d) 8
25. 25.What is the units digit of 8^41? (a) 6 (b) 4 (c) 2 (d) 8
26. 26.Which of these numbers is divisible by 3? (a) 675 (b) 674 (c) 676 (d) 679
27. 27.What is the units digit of 9^57? (a) 1 (b) 8 (c) 9 (d) 7
28. 28.Which of these numbers is divisible by 6? (a) 2074 (b) 2069 (c) 2063 (d) 2064
29. 29.Which of these numbers is divisible by 9? (a) 1431 (b) 1424 (c) 1436 (d) 1445
30. 30.Which of these numbers is divisible by 4? (a) 3565 (b) 3569 (c) 3568 (d) 3573
31. 31.Which of these numbers is divisible by 3? (a) 2153 (b) 2147 (c) 2150 (d) 2148
32. 32.Which of these numbers is divisible by 8? (a) 6501 (b) 6509 (c) 6513 (d) 6504
33. 33.What is the units digit of 7^15? (a) 1 (b) 9 (c) 7 (d) 3
34. 34.What is the smallest number you must add to 308 to make it divisible by 5? (a) 1 (b) 4 (c) 2 (d) 3
35. 35.What is the remainder when 731 is divided by 11? (a) 6 (b) 7 (c) 4 (d) 5
36. 36.Which of these numbers is divisible by 11? (a) 7152 (b) 7148 (c) 7139 (d) 7133
37. 37.Which of these numbers is divisible by 6? (a) 779 (b) 774 (c) 784 (d) 769
38. 38.Which of these numbers is divisible by 4? (a) 1203 (b) 1196 (c) 1194 (d) 1198
39. 39.Which of these numbers is divisible by 4? (a) 518 (b) 512 (c) 510 (d) 513
40. 40.Which of these numbers is divisible by 9? (a) 5257 (b) 5249 (c) 5256 (d) 5272

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