Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 883 · 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 6? (a) 5237 (b) 5232 (c) 5241 (d) 5231
2. 2.What is the smallest number you must add to 147 to make it divisible by 4? (a) 2 (b) 3 (c) 1
3. 3.What is the remainder when 367 is divided by 10? (a) 6 (b) 3 (c) 8 (d) 7
4. 4.Which of these numbers is divisible by 8? (a) 1650 (b) 1661 (c) 1665 (d) 1656
5. 5.What is the remainder when 643 is divided by 12? (a) 8 (b) 5 (c) 6 (d) 7
6. 6.What is the remainder when 107 is divided by 9? (a) 6 (b) 1 (c) 8 (d) 7
7. 7.Which of these numbers is divisible by 4? (a) 2998 (b) 2993 (c) 2992 (d) 2989
8. 8.What is the units digit of 3^42? (a) 3 (b) 7 (c) 9 (d) 1
9. 9.What is the smallest number you must add to 730 to make it divisible by 12? (a) 3 (b) 1 (c) 10 (d) 2
10. 10.What is the smallest number you must add to 73 to make it divisible by 4? (a) 1 (b) 3 (c) 2
11. 11.What is the remainder when 258 is divided by 9? (a) 5 (b) 7 (c) 6 (d) 3
12. 12.What is the smallest number you must add to 545 to make it divisible by 8? (a) 1 (b) 6 (c) 7 (d) 5
13. 13.Which of these numbers is divisible by 9? (a) 7038 (b) 7031 (c) 7045 (d) 7055
14. 14.What is the smallest number you must add to 469 to make it divisible by 11? (a) 5 (b) 4 (c) 7 (d) 3
15. 15.Which of these numbers is divisible by 6? (a) 2250 (b) 2259 (c) 2252 (d) 2246
16. 16.What is the units digit of 3^40? (a) 3 (b) 9 (c) 7 (d) 1
17. 17.What is the remainder when 118 is divided by 7? (a) 5 (b) 4 (c) 1 (d) 6
18. 18.What is the remainder when 179 is divided by 3? (a) 0 (b) 1 (c) 2
19. 19.What is the units digit of 9^18? (a) 9 (b) 0 (c) 2 (d) 1
20. 20.What is the remainder when 530 is divided by 7? (a) 4 (b) 6 (c) 5 (d) 2
21. 21.What is the units digit of 2^27? (a) 2 (b) 6 (c) 4 (d) 8
22. 22.What is the remainder when 241 is divided by 8? (a) 0 (b) 2 (c) 7 (d) 1
23. 23.What is the smallest number you must add to 103 to make it divisible by 9? (a) 6 (b) 7 (c) 4 (d) 5
24. 24.Which of these numbers is divisible by 9? (a) 847 (b) 835 (c) 837 (d) 842
25. 25.Which of these numbers is divisible by 8? (a) 5823 (b) 5816 (c) 5810 (d) 5830
26. 26.What is the smallest number you must add to 401 to make it divisible by 7? (a) 5 (b) 4 (c) 6 (d) 2
27. 27.Which of these numbers is divisible by 9? (a) 4921 (b) 4913 (c) 4905 (d) 4898
28. 28.What is the remainder when 403 is divided by 8? (a) 5 (b) 4 (c) 2 (d) 3
29. 29.Which of these numbers is divisible by 3? (a) 1665 (b) 1670 (c) 1663 (d) 1667
30. 30.What is the smallest number you must add to 238 to make it divisible by 9? (a) 5 (b) 7 (c) 4 (d) 6
31. 31.What is the remainder when 685 is divided by 10? (a) 4 (b) 6 (c) 5 (d) 7
32. 32.What is the smallest number you must add to 627 to make it divisible by 12? (a) 3 (b) 9 (c) 10 (d) 8
33. 33.Which of these numbers is divisible by 6? (a) 680 (b) 684 (c) 693 (d) 687
34. 34.What is the remainder when 576 is divided by 10? (a) 6 (b) 7 (c) 4 (d) 5
35. 35.Which of these numbers is divisible by 3? (a) 1887 (b) 1892 (c) 1889 (d) 1886
36. 36.What is the units digit of 4^22? (a) 7 (b) 4 (c) 5 (d) 6
37. 37.Which of these numbers is divisible by 6? (a) 5379 (b) 5374 (c) 5383 (d) 5376
38. 38.What is the remainder when 375 is divided by 6? (a) 5 (b) 4 (c) 2 (d) 3
39. 39.What is the remainder when 94 is divided by 3? (a) 1 (b) 2 (c) 0
40. 40.What is the smallest number you must add to 762 to make it divisible by 12? (a) 8 (b) 6 (c) 5 (d) 7

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