Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 806 · 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 561 to make it divisible by 7? (a) 1 (b) 5 (c) 6 (d) 4
2. 2.What is the units digit of 9^57? (a) 9 (b) 1 (c) 8 (d) 7
3. 3.What is the units digit of 7^20? (a) 1 (b) 3 (c) 9 (d) 7
4. 4.What is the smallest number you must add to 81 to make it divisible by 6? (a) 2 (b) 5 (c) 3 (d) 4
5. 5.What is the remainder when 109 is divided by 6? (a) 1 (b) 5 (c) 0 (d) 2
6. 6.What is the units digit of 3^19? (a) 3 (b) 7 (c) 1 (d) 9
7. 7.What is the smallest number you must add to 319 to make it divisible by 5? (a) 4 (b) 1 (c) 2 (d) 3
8. 8.Which of these numbers is divisible by 4? (a) 774 (b) 772 (c) 770 (d) 777
9. 9.Which of these numbers is divisible by 11? (a) 3825 (b) 3806 (c) 3807 (d) 3799
10. 10.Which of these numbers is divisible by 6? (a) 4988 (b) 4981 (c) 4980 (d) 4979
11. 11.Which of these numbers is divisible by 11? (a) 3808 (b) 3797 (c) 3787 (d) 3795
12. 12.What is the smallest number you must add to 204 to make it divisible by 5? (a) 2 (b) 1 (c) 4 (d) 3
13. 13.What is the remainder when 133 is divided by 5? (a) 1 (b) 2 (c) 3 (d) 4
14. 14.Which of these numbers is divisible by 11? (a) 976 (b) 981 (c) 979 (d) 996
15. 15.Which of these numbers is divisible by 6? (a) 2255 (b) 2246 (c) 2242 (d) 2244
16. 16.What is the units digit of 2^32? (a) 6 (b) 2 (c) 8 (d) 4
17. 17.Which of these numbers is divisible by 11? (a) 790 (b) 770 (c) 766 (d) 773
18. 18.What is the remainder when 205 is divided by 11? (a) 4 (b) 8 (c) 7 (d) 6
19. 19.What is the units digit of 2^41? (a) 2 (b) 4 (c) 6 (d) 8
20. 20.Which of these numbers is divisible by 11? (a) 3761 (b) 3742 (c) 3730 (d) 3740
21. 21.What is the remainder when 98 is divided by 3? (a) 1 (b) 2 (c) 0
22. 22.What is the units digit of 3^17? (a) 3 (b) 9 (c) 7 (d) 1
23. 23.What is the units digit of 2^14? (a) 2 (b) 4 (c) 6 (d) 8
24. 24.What is the remainder when 146 is divided by 5? (a) 0 (b) 1 (c) 2 (d) 4
25. 25.What is the remainder when 386 is divided by 5? (a) 1 (b) 2 (c) 0 (d) 4
26. 26.What is the units digit of 8^37? (a) 2 (b) 6 (c) 4 (d) 8
27. 27.What is the smallest number you must add to 310 to make it divisible by 11? (a) 2 (b) 8 (c) 10 (d) 9
28. 28.What is the units digit of 2^45? (a) 8 (b) 4 (c) 2 (d) 6
29. 29.What is the remainder when 252 is divided by 11? (a) 9 (b) 8 (c) 1 (d) 10
30. 30.What is the remainder when 168 is divided by 9? (a) 7 (b) 6 (c) 3 (d) 5
31. 31.Which of these numbers is divisible by 9? (a) 2565 (b) 2575 (c) 2566 (d) 2563
32. 32.What is the remainder when 246 is divided by 11? (a) 4 (b) 3 (c) 7 (d) 5
33. 33.What is the remainder when 202 is divided by 9? (a) 3 (b) 5 (c) 6 (d) 4
34. 34.Which of these numbers is divisible by 3? (a) 1428 (b) 1433 (c) 1427 (d) 1429
35. 35.What is the smallest number you must add to 159 to make it divisible by 8? (a) 1 (b) 3 (c) 2 (d) 7
36. 36.Which of these numbers is divisible by 11? (a) 5410 (b) 5390 (c) 5385 (d) 5395
37. 37.What is the smallest number you must add to 137 to make it divisible by 10? (a) 4 (b) 2 (c) 3 (d) 7
38. 38.What is the units digit of 8^52? (a) 4 (b) 6 (c) 8 (d) 2
39. 39.What is the smallest number you must add to 199 to make it divisible by 5? (a) 4 (b) 1 (c) 3 (d) 2
40. 40.What is the smallest number you must add to 43 to make it divisible by 3? (a) 1 (b) 2

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