Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 2 · 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 631 is divided by 9? (a) 8 (b) 2 (c) 0 (d) 1
2. 2.Which of these numbers is divisible by 4? (a) 223 (b) 217 (c) 225 (d) 220
3. 3.What is the smallest number you must add to 304 to make it divisible by 9? (a) 7 (b) 2 (c) 1 (d) 3
4. 4.Which of these numbers is divisible by 6? (a) 2963 (b) 2974 (c) 2964 (d) 2967
5. 5.What is the remainder when 285 is divided by 7? (a) 2 (b) 5 (c) 6 (d) 4
6. 6.What is the smallest number you must add to 308 to make it divisible by 5? (a) 3 (b) 4 (c) 1 (d) 2
7. 7.Which of these numbers is divisible by 8? (a) 998 (b) 979 (c) 984 (d) 986
8. 8.Which of these numbers is divisible by 6? (a) 3042 (b) 3040 (c) 3043 (d) 3051
9. 9.Which of these numbers is divisible by 3? (a) 731 (b) 736 (c) 732 (d) 734
10. 10.What is the remainder when 251 is divided by 4? (a) 2 (b) 1 (c) 0 (d) 3
11. 11.Which of these numbers is divisible by 8? (a) 2637 (b) 2627 (c) 2632 (d) 2645
12. 12.What is the remainder when 261 is divided by 4? (a) 2 (b) 0 (c) 1 (d) 3
13. 13.What is the smallest number you must add to 456 to make it divisible by 9? (a) 4 (b) 6 (c) 2 (d) 3
14. 14.What is the remainder when 322 is divided by 5? (a) 1 (b) 2 (c) 3 (d) 4
15. 15.What is the units digit of 9^15? (a) 1 (b) 8 (c) 7 (d) 9
16. 16.What is the remainder when 143 is divided by 5? (a) 4 (b) 2 (c) 1 (d) 3
17. 17.Which of these numbers is divisible by 4? (a) 1617 (b) 1622 (c) 1625 (d) 1620
18. 18.What is the remainder when 178 is divided by 3? (a) 2 (b) 0 (c) 1
19. 19.What is the remainder when 79 is divided by 4? (a) 0 (b) 1 (c) 2 (d) 3
20. 20.What is the remainder when 671 is divided by 9? (a) 7 (b) 5 (c) 6 (d) 4
21. 21.What is the remainder when 397 is divided by 5? (a) 4 (b) 3 (c) 2 (d) 1
22. 22.What is the smallest number you must add to 267 to make it divisible by 6? (a) 3 (b) 2 (c) 4 (d) 5
23. 23.What is the smallest number you must add to 134 to make it divisible by 3? (a) 2 (b) 1
24. 24.What is the smallest number you must add to 139 to make it divisible by 3? (a) 2 (b) 1
25. 25.Which of these numbers is divisible by 6? (a) 1005 (b) 1011 (c) 1008 (d) 1015
26. 26.What is the units digit of 9^19? (a) 9 (b) 7 (c) 8 (d) 1
27. 27.What is the remainder when 234 is divided by 8? (a) 1 (b) 3 (c) 6 (d) 2
28. 28.What is the remainder when 406 is divided by 8? (a) 7 (b) 5 (c) 2 (d) 6
29. 29.Which of these numbers is divisible by 4? (a) 3356 (b) 3357 (c) 3361 (d) 3353
30. 30.Which of these numbers is divisible by 3? (a) 841 (b) 848 (c) 844 (d) 843
31. 31.What is the units digit of 3^22? (a) 3 (b) 1 (c) 7 (d) 9
32. 32.What is the smallest number you must add to 467 to make it divisible by 8? (a) 3 (b) 4 (c) 5 (d) 6
33. 33.Which of these numbers is divisible by 9? (a) 5144 (b) 5158 (c) 5148 (d) 5154
34. 34.What is the smallest number you must add to 391 to make it divisible by 10? (a) 1 (b) 9 (c) 8 (d) 7
35. 35.What is the smallest number you must add to 103 to make it divisible by 6? (a) 1 (b) 4 (c) 5 (d) 3
36. 36.Which of these numbers is divisible by 8? (a) 2065 (b) 2057 (c) 2076 (d) 2064
37. 37.Which of these numbers is divisible by 6? (a) 1051 (b) 1057 (c) 1050 (d) 1046
38. 38.Which of these numbers is divisible by 11? (a) 6002 (b) 6011 (c) 5992 (d) 5995
39. 39.What is the smallest number you must add to 699 to make it divisible by 11? (a) 6 (b) 7 (c) 4 (d) 5
40. 40.What is the remainder when 162 is divided by 12? (a) 7 (b) 5 (c) 6 (d) 8

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