Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 240 · 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 140 to make it divisible by 3? (a) 1 (b) 2
2. 2.Which of these numbers is divisible by 3? (a) 387 (b) 389 (c) 386 (d) 391
3. 3.What is the units digit of 8^34? (a) 2 (b) 8 (c) 6 (d) 4
4. 4.What is the remainder when 542 is divided by 11? (a) 2 (b) 8 (c) 4 (d) 3
5. 5.What is the remainder when 401 is divided by 5? (a) 2 (b) 0 (c) 1 (d) 4
6. 6.Which of these numbers is divisible by 8? (a) 1235 (b) 1232 (c) 1230 (d) 1244
7. 7.What is the units digit of 9^42? (a) 1 (b) 2 (c) 0 (d) 9
8. 8.What is the smallest number you must add to 95 to make it divisible by 3? (a) 1 (b) 2
9. 9.What is the units digit of 9^56? (a) 2 (b) 1 (c) 9 (d) 0
10. 10.What is the units digit of 3^19? (a) 1 (b) 3 (c) 7 (d) 9
11. 11.What is the units digit of 7^14? (a) 9 (b) 1 (c) 3 (d) 7
12. 12.What is the smallest number you must add to 279 to make it divisible by 8? (a) 7 (b) 1 (c) 3 (d) 2
13. 13.Which of these numbers is divisible by 9? (a) 667 (b) 656 (c) 665 (d) 657
14. 14.What is the smallest number you must add to 254 to make it divisible by 4? (a) 3 (b) 1 (c) 2
15. 15.What is the units digit of 9^32? (a) 0 (b) 2 (c) 1 (d) 9
16. 16.Which of these numbers is divisible by 6? (a) 2771 (b) 2779 (c) 2772 (d) 2777
17. 17.What is the units digit of 2^56? (a) 2 (b) 4 (c) 6 (d) 8
18. 18.Which of these numbers is divisible by 3? (a) 548 (b) 549 (c) 550 (d) 553
19. 19.What is the remainder when 168 is divided by 11? (a) 8 (b) 4 (c) 2 (d) 3
20. 20.Which of these numbers is divisible by 3? (a) 1136 (b) 1132 (c) 1138 (d) 1134
21. 21.What is the smallest number you must add to 508 to make it divisible by 9? (a) 7 (b) 4 (c) 6 (d) 5
22. 22.Which of these numbers is divisible by 9? (a) 4548 (b) 4558 (c) 4538 (d) 4545
23. 23.What is the remainder when 611 is divided by 8? (a) 3 (b) 2 (c) 4 (d) 5
24. 24.What is the remainder when 398 is divided by 5? (a) 4 (b) 1 (c) 3 (d) 2
25. 25.What is the units digit of 7^37? (a) 7 (b) 9 (c) 3 (d) 1
26. 26.What is the smallest number you must add to 366 to make it divisible by 5? (a) 1 (b) 2 (c) 4 (d) 3
27. 27.What is the units digit of 9^44? (a) 9 (b) 2 (c) 0 (d) 1
28. 28.What is the units digit of 2^41? (a) 4 (b) 8 (c) 2 (d) 6
29. 29.Which of these numbers is divisible by 11? (a) 4613 (b) 4606 (c) 4623 (d) 4609
30. 30.What is the remainder when 43 is divided by 3? (a) 0 (b) 1 (c) 2
31. 31.What is the units digit of 3^41? (a) 1 (b) 3 (c) 7 (d) 9
32. 32.Which of these numbers is divisible by 6? (a) 3470 (b) 3478 (c) 3481 (d) 3474
33. 33.What is the smallest number you must add to 106 to make it divisible by 6? (a) 3 (b) 2 (c) 1 (d) 4
34. 34.Which of these numbers is divisible by 4? (a) 1889 (b) 1895 (c) 1892 (d) 1899
35. 35.What is the smallest number you must add to 532 to make it divisible by 9? (a) 1 (b) 8 (c) 6 (d) 7
36. 36.Which of these numbers is divisible by 3? (a) 1903 (b) 1899 (c) 1898 (d) 1900
37. 37.What is the smallest number you must add to 195 to make it divisible by 9? (a) 4 (b) 2 (c) 6 (d) 3
38. 38.What is the units digit of 9^43? (a) 8 (b) 7 (c) 9 (d) 1
39. 39.Which of these numbers is divisible by 3? (a) 1682 (b) 1679 (c) 1677 (d) 1676
40. 40.What is the smallest number you must add to 628 to make it divisible by 10? (a) 3 (b) 2 (c) 8 (d) 1

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