Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 718 · 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 6? (a) 1223 (b) 1208 (c) 1212 (d) 1217
2. 2.What is the units digit of 3^43? (a) 1 (b) 7 (c) 3 (d) 9
3. 3.What is the units digit of 2^25? (a) 4 (b) 2 (c) 6 (d) 8
4. 4.What is the remainder when 533 is divided by 9? (a) 3 (b) 1 (c) 7 (d) 2
5. 5.Which of these numbers is divisible by 8? (a) 5733 (b) 5720 (c) 5715 (d) 5726
6. 6.What is the smallest number you must add to 63 to make it divisible by 6? (a) 4 (b) 5 (c) 3 (d) 2
7. 7.What is the units digit of 4^34? (a) 4 (b) 5 (c) 6 (d) 7
8. 8.Which of these numbers is divisible by 4? (a) 2315 (b) 2316 (c) 2323 (d) 2319
9. 9.Which of these numbers is divisible by 11? (a) 8644 (b) 8630 (c) 8651 (d) 8635
10. 10.Which of these numbers is divisible by 3? (a) 526 (b) 530 (c) 524 (d) 525
11. 11.What is the units digit of 9^30? (a) 9 (b) 0 (c) 1 (d) 2
12. 12.Which of these numbers is divisible by 4? (a) 522 (b) 517 (c) 527 (d) 520
13. 13.What is the units digit of 3^16? (a) 3 (b) 9 (c) 7 (d) 1
14. 14.Which of these numbers is divisible by 8? (a) 6519 (b) 6512 (c) 6525 (d) 6509
15. 15.What is the units digit of 4^52? (a) 6 (b) 7 (c) 5 (d) 4
16. 16.What is the smallest number you must add to 32 to make it divisible by 3? (a) 1 (b) 2
17. 17.What is the remainder when 289 is divided by 7? (a) 3 (b) 1 (c) 5 (d) 2
18. 18.Which of these numbers is divisible by 6? (a) 1686 (b) 1683 (c) 1687 (d) 1697
19. 19.What is the remainder when 507 is divided by 7? (a) 2 (b) 4 (c) 5 (d) 3
20. 20.What is the units digit of 7^50? (a) 7 (b) 1 (c) 9 (d) 3
21. 21.What is the smallest number you must add to 92 to make it divisible by 6? (a) 4 (b) 2 (c) 5 (d) 3
22. 22.What is the remainder when 534 is divided by 7? (a) 5 (b) 1 (c) 3 (d) 2
23. 23.What is the remainder when 255 is divided by 12? (a) 4 (b) 2 (c) 3 (d) 9
24. 24.What is the units digit of 2^13? (a) 8 (b) 6 (c) 2 (d) 4
25. 25.Which of these numbers is divisible by 11? (a) 9867 (b) 9857 (c) 9885 (d) 9872
26. 26.What is the remainder when 173 is divided by 5? (a) 2 (b) 1 (c) 4 (d) 3
27. 27.Which of these numbers is divisible by 3? (a) 1196 (b) 1191 (c) 1190 (d) 1193
28. 28.What is the units digit of 3^36? (a) 9 (b) 7 (c) 1 (d) 3
29. 29.Which of these numbers is divisible by 11? (a) 7469 (b) 7475 (c) 7482 (d) 7462
30. 30.What is the remainder when 77 is divided by 6? (a) 3 (b) 4 (c) 1 (d) 5
31. 31.What is the smallest number you must add to 179 to make it divisible by 3? (a) 2 (b) 1
32. 32.What is the units digit of 3^10? (a) 1 (b) 3 (c) 7 (d) 9
33. 33.What is the smallest number you must add to 68 to make it divisible by 3? (a) 2 (b) 1
34. 34.What is the remainder when 302 is divided by 9? (a) 5 (b) 6 (c) 4 (d) 7
35. 35.Which of these numbers is divisible by 8? (a) 1944 (b) 1955 (c) 1937 (d) 1945
36. 36.Which of these numbers is divisible by 11? (a) 3299 (b) 3278 (c) 3270 (d) 3281
37. 37.Which of these numbers is divisible by 4? (a) 570 (b) 573 (c) 578 (d) 572
38. 38.What is the remainder when 228 is divided by 5? (a) 2 (b) 1 (c) 3 (d) 4
39. 39.What is the smallest number you must add to 146 to make it divisible by 9? (a) 8 (b) 2 (c) 6 (d) 7
40. 40.Which of these numbers is divisible by 9? (a) 1990 (b) 2015 (c) 1998 (d) 2002

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