Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 234 · 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 728 to make it divisible by 9? (a) 3 (b) 2 (c) 1 (d) 8
2. 2.What is the smallest number you must add to 582 to make it divisible by 8? (a) 2 (b) 3 (c) 1 (d) 6
3. 3.What is the remainder when 97 is divided by 5? (a) 1 (b) 4 (c) 2 (d) 3
4. 4.What is the units digit of 7^34? (a) 1 (b) 9 (c) 7 (d) 3
5. 5.What is the units digit of 7^14? (a) 7 (b) 1 (c) 3 (d) 9
6. 6.Which of these numbers is divisible by 11? (a) 4410 (b) 4392 (c) 4416 (d) 4400
7. 7.What is the smallest number you must add to 214 to make it divisible by 3? (a) 2 (b) 1
8. 8.What is the remainder when 287 is divided by 10? (a) 3 (b) 8 (c) 6 (d) 7
9. 9.What is the smallest number you must add to 154 to make it divisible by 6? (a) 3 (b) 4 (c) 1 (d) 2
10. 10.What is the remainder when 684 is divided by 11? (a) 9 (b) 3 (c) 1 (d) 2
11. 11.What is the remainder when 237 is divided by 4? (a) 3 (b) 0 (c) 2 (d) 1
12. 12.Which of these numbers is divisible by 11? (a) 1611 (b) 1634 (c) 1617 (d) 1623
13. 13.Which of these numbers is divisible by 8? (a) 2417 (b) 2410 (c) 2416 (d) 2427
14. 14.Which of these numbers is divisible by 11? (a) 3165 (b) 3187 (c) 3168 (d) 3172
15. 15.Which of these numbers is divisible by 9? (a) 4648 (b) 4637 (c) 4631 (d) 4635
16. 16.What is the units digit of 4^23? (a) 6 (b) 5 (c) 4 (d) 3
17. 17.Which of these numbers is divisible by 8? (a) 6765 (b) 6752 (c) 6749 (d) 6753
18. 18.What is the smallest number you must add to 157 to make it divisible by 6? (a) 4 (b) 3 (c) 1 (d) 5
19. 19.What is the smallest number you must add to 230 to make it divisible by 4? (a) 2 (b) 3 (c) 1
20. 20.Which of these numbers is divisible by 8? (a) 682 (b) 694 (c) 679 (d) 680
21. 21.What is the remainder when 35 is divided by 3? (a) 2 (b) 1 (c) 0
22. 22.Which of these numbers is divisible by 8? (a) 6553 (b) 6544 (c) 6537 (d) 6550
23. 23.What is the units digit of 7^47? (a) 1 (b) 9 (c) 3 (d) 7
24. 24.What is the units digit of 4^38? (a) 4 (b) 7 (c) 6 (d) 5
25. 25.What is the units digit of 4^21? (a) 5 (b) 4 (c) 3 (d) 6
26. 26.What is the remainder when 158 is divided by 7? (a) 3 (b) 4 (c) 6 (d) 5
27. 27.Which of these numbers is divisible by 3? (a) 1076 (b) 1070 (c) 1072 (d) 1071
28. 28.What is the units digit of 7^33? (a) 3 (b) 7 (c) 1 (d) 9
29. 29.What is the units digit of 4^33? (a) 5 (b) 6 (c) 4 (d) 3
30. 30.What is the units digit of 8^53? (a) 4 (b) 2 (c) 6 (d) 8
31. 31.Which of these numbers is divisible by 9? (a) 7896 (b) 7893 (c) 7890 (d) 7909
32. 32.What is the remainder when 239 is divided by 12? (a) 11 (b) 10 (c) 9 (d) 1
33. 33.What is the remainder when 474 is divided by 12? (a) 7 (b) 5 (c) 8 (d) 6
34. 34.What is the units digit of 9^13? (a) 1 (b) 9 (c) 8 (d) 7
35. 35.What is the units digit of 8^50? (a) 8 (b) 2 (c) 6 (d) 4
36. 36.What is the remainder when 146 is divided by 12? (a) 3 (b) 2 (c) 1 (d) 10
37. 37.What is the units digit of 7^19? (a) 7 (b) 3 (c) 1 (d) 9
38. 38.What is the units digit of 7^29? (a) 7 (b) 1 (c) 9 (d) 3
39. 39.Which of these numbers is divisible by 3? (a) 462 (b) 460 (c) 467 (d) 463
40. 40.Which of these numbers is divisible by 4? (a) 3535 (b) 3538 (c) 3541 (d) 3536

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