Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 241 · 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 9^52? (a) 2 (b) 0 (c) 1 (d) 9
2. 2.Which of these numbers is divisible by 4? (a) 456 (b) 454 (c) 463 (d) 458
3. 3.What is the units digit of 2^25? (a) 8 (b) 4 (c) 6 (d) 2
4. 4.What is the units digit of 4^21? (a) 5 (b) 4 (c) 6 (d) 3
5. 5.What is the smallest number you must add to 194 to make it divisible by 12? (a) 11 (b) 2 (c) 10 (d) 9
6. 6.What is the smallest number you must add to 227 to make it divisible by 5? (a) 1 (b) 4 (c) 2 (d) 3
7. 7.Which of these numbers is divisible by 6? (a) 5251 (b) 5250 (c) 5260 (d) 5246
8. 8.Which of these numbers is divisible by 8? (a) 4747 (b) 4744 (c) 4756 (d) 4741
9. 9.What is the remainder when 383 is divided by 11? (a) 10 (b) 2 (c) 9 (d) 8
10. 10.What is the smallest number you must add to 558 to make it divisible by 8? (a) 6 (b) 3 (c) 1 (d) 2
11. 11.What is the remainder when 472 is divided by 7? (a) 4 (b) 3 (c) 5 (d) 2
12. 12.What is the remainder when 305 is divided by 4? (a) 2 (b) 1 (c) 3 (d) 0
13. 13.What is the remainder when 542 is divided by 7? (a) 2 (b) 3 (c) 5 (d) 4
14. 14.What is the remainder when 131 is divided by 3? (a) 2 (b) 0 (c) 1
15. 15.Which of these numbers is divisible by 8? (a) 6455 (b) 6444 (c) 6448 (d) 6463
16. 16.What is the remainder when 166 is divided by 4? (a) 3 (b) 2 (c) 0 (d) 1
17. 17.What is the remainder when 595 is divided by 8? (a) 2 (b) 3 (c) 4 (d) 5
18. 18.Which of these numbers is divisible by 4? (a) 1985 (b) 1980 (c) 1979 (d) 1983
19. 19.What is the remainder when 741 is divided by 10? (a) 0 (b) 9 (c) 1 (d) 2
20. 20.What is the units digit of 7^39? (a) 1 (b) 9 (c) 7 (d) 3
21. 21.Which of these numbers is divisible by 3? (a) 785 (b) 791 (c) 788 (d) 786
22. 22.What is the smallest number you must add to 80 to make it divisible by 3? (a) 2 (b) 1
23. 23.Which of these numbers is divisible by 4? (a) 2422 (b) 2426 (c) 2424 (d) 2429
24. 24.What is the remainder when 561 is divided by 9? (a) 3 (b) 6 (c) 2 (d) 4
25. 25.What is the remainder when 636 is divided by 8? (a) 5 (b) 6 (c) 3 (d) 4
26. 26.What is the remainder when 232 is divided by 7? (a) 0 (b) 1 (c) 2 (d) 6
27. 27.What is the remainder when 121 is divided by 3? (a) 0 (b) 1 (c) 2
28. 28.Which of these numbers is divisible by 3? (a) 290 (b) 286 (c) 292 (d) 288
29. 29.What is the smallest number you must add to 246 to make it divisible by 7? (a) 6 (b) 5 (c) 4 (d) 1
30. 30.What is the remainder when 307 is divided by 6? (a) 5 (b) 0 (c) 1 (d) 2
31. 31.What is the units digit of 3^22? (a) 7 (b) 9 (c) 1 (d) 3
32. 32.What is the units digit of 2^50? (a) 8 (b) 6 (c) 4 (d) 2
33. 33.What is the units digit of 3^35? (a) 3 (b) 9 (c) 7 (d) 1
34. 34.What is the smallest number you must add to 269 to make it divisible by 9? (a) 2 (b) 8 (c) 1 (d) 3
35. 35.What is the units digit of 8^26? (a) 8 (b) 2 (c) 6 (d) 4
36. 36.What is the smallest number you must add to 638 to make it divisible by 12? (a) 2 (b) 11 (c) 10 (d) 9
37. 37.What is the remainder when 625 is divided by 9? (a) 6 (b) 4 (c) 5 (d) 3
38. 38.Which of these numbers is divisible by 3? (a) 2248 (b) 2251 (c) 2255 (d) 2250
39. 39.Which of these numbers is divisible by 3? (a) 146 (b) 151 (c) 148 (d) 147
40. 40.Which of these numbers is divisible by 6? (a) 3526 (b) 3512 (c) 3519 (d) 3516

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