Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 312 · 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 units digit of 3^28? (a) 1 (b) 7 (c) 3 (d) 9
2. 2.Which of these numbers is divisible by 4? (a) 1675 (b) 1672 (c) 1671 (d) 1678
3. 3.Which of these numbers is divisible by 6? (a) 3788 (b) 3785 (c) 3780 (d) 3777
4. 4.What is the remainder when 175 is divided by 4? (a) 1 (b) 2 (c) 0 (d) 3
5. 5.What is the remainder when 109 is divided by 9? (a) 0 (b) 1 (c) 2 (d) 8
6. 6.Which of these numbers is divisible by 9? (a) 1760 (b) 1753 (c) 1766 (d) 1755
7. 7.What is the smallest number you must add to 44 to make it divisible by 3? (a) 1 (b) 2
8. 8.Which of these numbers is divisible by 8? (a) 911 (b) 917 (c) 902 (d) 904
9. 9.What is the remainder when 81 is divided by 6? (a) 3 (b) 5 (c) 2 (d) 4
10. 10.What is the remainder when 134 is divided by 7? (a) 0 (b) 2 (c) 6 (d) 1
11. 11.What is the units digit of 7^60? (a) 9 (b) 1 (c) 3 (d) 7
12. 12.What is the smallest number you must add to 157 to make it divisible by 7? (a) 6 (b) 5 (c) 3 (d) 4
13. 13.What is the remainder when 192 is divided by 7? (a) 3 (b) 2 (c) 5 (d) 4
14. 14.What is the smallest number you must add to 472 to make it divisible by 10? (a) 9 (b) 2 (c) 7 (d) 8
15. 15.What is the smallest number you must add to 654 to make it divisible by 12? (a) 5 (b) 7 (c) 6 (d) 8
16. 16.What is the smallest number you must add to 596 to make it divisible by 11? (a) 2 (b) 8 (c) 9 (d) 10
17. 17.What is the units digit of 2^58? (a) 8 (b) 2 (c) 6 (d) 4
18. 18.Which of these numbers is divisible by 9? (a) 7147 (b) 7137 (c) 7138 (d) 7134
19. 19.What is the units digit of 2^60? (a) 2 (b) 8 (c) 4 (d) 6
20. 20.What is the units digit of 9^38? (a) 2 (b) 1 (c) 0 (d) 9
21. 21.What is the smallest number you must add to 133 to make it divisible by 5? (a) 2 (b) 1 (c) 4 (d) 3
22. 22.Which of these numbers is divisible by 3? (a) 2648 (b) 2651 (c) 2644 (d) 2646
23. 23.Which of these numbers is divisible by 11? (a) 7898 (b) 7897 (c) 7914 (d) 7899
24. 24.Which of these numbers is divisible by 6? (a) 855 (b) 846 (c) 845 (d) 851
25. 25.What is the smallest number you must add to 441 to make it divisible by 8? (a) 1 (b) 6 (c) 5 (d) 7
26. 26.Which of these numbers is divisible by 3? (a) 1969 (b) 1967 (c) 1972 (d) 1968
27. 27.What is the units digit of 4^25? (a) 4 (b) 3 (c) 6 (d) 5
28. 28.What is the smallest number you must add to 235 to make it divisible by 9? (a) 1 (b) 8 (c) 6 (d) 7
29. 29.What is the remainder when 157 is divided by 3? (a) 1 (b) 0 (c) 2
30. 30.What is the units digit of 3^17? (a) 3 (b) 1 (c) 7 (d) 9
31. 31.What is the units digit of 9^21? (a) 9 (b) 1 (c) 8 (d) 7
32. 32.What is the smallest number you must add to 170 to make it divisible by 11? (a) 7 (b) 5 (c) 8 (d) 6
33. 33.Which of these numbers is divisible by 9? (a) 4481 (b) 4488 (c) 4473 (d) 4466
34. 34.What is the units digit of 7^52? (a) 7 (b) 9 (c) 3 (d) 1
35. 35.What is the smallest number you must add to 488 to make it divisible by 12? (a) 8 (b) 3 (c) 5 (d) 4
36. 36.What is the smallest number you must add to 257 to make it divisible by 4? (a) 2 (b) 1 (c) 3
37. 37.What is the smallest number you must add to 199 to make it divisible by 8? (a) 2 (b) 7 (c) 3 (d) 1
38. 38.Which of these numbers is divisible by 9? (a) 2664 (b) 2680 (c) 2668 (d) 2662
39. 39.What is the remainder when 118 is divided by 5? (a) 4 (b) 3 (c) 1 (d) 2
40. 40.What is the remainder when 299 is divided by 11? (a) 3 (b) 9 (c) 2 (d) 1

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