Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 710 · 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 11? (a) 1642 (b) 1630 (c) 1639 (d) 1660
2. 2.What is the smallest number you must add to 619 to make it divisible by 12? (a) 4 (b) 5 (c) 7 (d) 6
3. 3.What is the remainder when 293 is divided by 5? (a) 1 (b) 2 (c) 4 (d) 3
4. 4.What is the units digit of 9^21? (a) 8 (b) 9 (c) 7 (d) 1
5. 5.What is the units digit of 8^37? (a) 6 (b) 4 (c) 2 (d) 8
6. 6.What is the smallest number you must add to 127 to make it divisible by 9? (a) 8 (b) 1 (c) 7 (d) 6
7. 7.Which of these numbers is divisible by 3? (a) 645 (b) 650 (c) 644 (d) 647
8. 8.What is the smallest number you must add to 658 to make it divisible by 11? (a) 9 (b) 3 (c) 2 (d) 1
9. 9.What is the units digit of 7^51? (a) 1 (b) 3 (c) 7 (d) 9
10. 10.What is the units digit of 9^47? (a) 9 (b) 8 (c) 1 (d) 7
11. 11.What is the remainder when 172 is divided by 3? (a) 1 (b) 2 (c) 0
12. 12.What is the units digit of 9^54? (a) 1 (b) 0 (c) 2 (d) 9
13. 13.Which of these numbers is divisible by 6? (a) 4862 (b) 4859 (c) 4854 (d) 4853
14. 14.Which of these numbers is divisible by 11? (a) 8791 (b) 8788 (c) 8778 (d) 8768
15. 15.Which of these numbers is divisible by 4? (a) 1888 (b) 1893 (c) 1889 (d) 1885
16. 16.What is the smallest number you must add to 688 to make it divisible by 12? (a) 4 (b) 7 (c) 8 (d) 9
17. 17.Which of these numbers is divisible by 3? (a) 2319 (b) 2320 (c) 2324 (d) 2318
18. 18.Which of these numbers is divisible by 3? (a) 549 (b) 548 (c) 554 (d) 551
19. 19.What is the units digit of 4^57? (a) 5 (b) 3 (c) 6 (d) 4
20. 20.What is the remainder when 609 is divided by 8? (a) 2 (b) 7 (c) 0 (d) 1
21. 21.What is the smallest number you must add to 386 to make it divisible by 12? (a) 10 (b) 11 (c) 2 (d) 9
22. 22.Which of these numbers is divisible by 11? (a) 6459 (b) 6471 (c) 6488 (d) 6468
23. 23.What is the units digit of 2^23? (a) 8 (b) 4 (c) 6 (d) 2
24. 24.Which of these numbers is divisible by 3? (a) 1331 (b) 1327 (c) 1324 (d) 1326
25. 25.What is the units digit of 3^35? (a) 1 (b) 9 (c) 7 (d) 3
26. 26.Which of these numbers is divisible by 3? (a) 920 (b) 918 (c) 917 (d) 923
27. 27.What is the smallest number you must add to 312 to make it divisible by 11? (a) 6 (b) 7 (c) 8 (d) 4
28. 28.What is the units digit of 3^11? (a) 3 (b) 9 (c) 7 (d) 1
29. 29.What is the smallest number you must add to 172 to make it divisible by 10? (a) 7 (b) 8 (c) 2 (d) 9
30. 30.Which of these numbers is divisible by 3? (a) 2227 (b) 2225 (c) 2230 (d) 2226
31. 31.Which of these numbers is divisible by 8? (a) 3287 (b) 3267 (c) 3272 (d) 3279
32. 32.Which of these numbers is divisible by 8? (a) 1553 (b) 1575 (c) 1567 (d) 1560
33. 33.What is the remainder when 358 is divided by 6? (a) 3 (b) 2 (c) 4 (d) 5
34. 34.Which of these numbers is divisible by 4? (a) 505 (b) 509 (c) 515 (d) 508
35. 35.What is the units digit of 4^42? (a) 6 (b) 7 (c) 4 (d) 5
36. 36.What is the remainder when 398 is divided by 11? (a) 1 (b) 9 (c) 3 (d) 2
37. 37.What is the smallest number you must add to 782 to make it divisible by 12? (a) 10 (b) 11 (c) 2 (d) 9
38. 38.Which of these numbers is divisible by 6? (a) 2849 (b) 2850 (c) 2853 (d) 2858
39. 39.What is the remainder when 353 is divided by 8? (a) 7 (b) 2 (c) 1 (d) 0
40. 40.What is the remainder when 855 is divided by 12? (a) 4 (b) 2 (c) 9 (d) 3

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