Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 367 · 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 3^60? (a) 9 (b) 1 (c) 3 (d) 7
2. 2.What is the remainder when 70 is divided by 4? (a) 0 (b) 1 (c) 3 (d) 2
3. 3.What is the remainder when 609 is divided by 10? (a) 1 (b) 7 (c) 9 (d) 8
4. 4.What is the units digit of 7^19? (a) 9 (b) 7 (c) 1 (d) 3
5. 5.Which of these numbers is divisible by 3? (a) 1813 (b) 1810 (c) 1812 (d) 1817
6. 6.What is the remainder when 293 is divided by 4? (a) 3 (b) 2 (c) 1 (d) 0
7. 7.Which of these numbers is divisible by 9? (a) 408 (b) 405 (c) 397 (d) 422
8. 8.What is the smallest number you must add to 82 to make it divisible by 3? (a) 1 (b) 2
9. 9.Which of these numbers is divisible by 8? (a) 582 (b) 568 (c) 574 (d) 564
10. 10.Which of these numbers is divisible by 6? (a) 3208 (b) 3221 (c) 3212 (d) 3210
11. 11.What is the smallest number you must add to 727 to make it divisible by 12? (a) 5 (b) 6 (c) 7 (d) 4
12. 12.What is the smallest number you must add to 206 to make it divisible by 12? (a) 10 (b) 2 (c) 11 (d) 9
13. 13.Which of these numbers is divisible by 4? (a) 688 (b) 695 (c) 686 (d) 690
14. 14.What is the units digit of 8^59? (a) 2 (b) 4 (c) 8 (d) 6
15. 15.Which of these numbers is divisible by 9? (a) 3409 (b) 3402 (c) 3397 (d) 3414
16. 16.What is the units digit of 7^18? (a) 7 (b) 9 (c) 1 (d) 3
17. 17.What is the units digit of 9^17? (a) 7 (b) 8 (c) 9 (d) 1
18. 18.Which of these numbers is divisible by 8? (a) 2819 (b) 2828 (c) 2816 (d) 2813
19. 19.What is the smallest number you must add to 295 to make it divisible by 4? (a) 1 (b) 3 (c) 2
20. 20.Which of these numbers is divisible by 4? (a) 3541 (b) 3538 (c) 3536 (d) 3535
21. 21.What is the smallest number you must add to 212 to make it divisible by 3? (a) 1 (b) 2
22. 22.What is the remainder when 196 is divided by 12? (a) 8 (b) 5 (c) 3 (d) 4
23. 23.What is the units digit of 4^44? (a) 5 (b) 4 (c) 7 (d) 6
24. 24.What is the remainder when 503 is divided by 12? (a) 1 (b) 10 (c) 9 (d) 11
25. 25.Which of these numbers is divisible by 6? (a) 3374 (b) 3364 (c) 3366 (d) 3367
26. 26.What is the smallest number you must add to 323 to make it divisible by 9? (a) 8 (b) 2 (c) 3 (d) 1
27. 27.What is the remainder when 431 is divided by 10? (a) 9 (b) 2 (c) 0 (d) 1
28. 28.What is the smallest number you must add to 341 to make it divisible by 7? (a) 1 (b) 5 (c) 3 (d) 2
29. 29.What is the remainder when 514 is divided by 8? (a) 1 (b) 3 (c) 2 (d) 6
30. 30.What is the remainder when 392 is divided by 6? (a) 4 (b) 2 (c) 1 (d) 3
31. 31.Which of these numbers is divisible by 6? (a) 4556 (b) 4554 (c) 4553 (d) 4561
32. 32.What is the remainder when 588 is divided by 11? (a) 7 (b) 4 (c) 6 (d) 5
33. 33.Which of these numbers is divisible by 4? (a) 1815 (b) 1805 (c) 1808 (d) 1811
34. 34.What is the units digit of 4^60? (a) 4 (b) 6 (c) 7 (d) 5
35. 35.What is the smallest number you must add to 219 to make it divisible by 4? (a) 2 (b) 1 (c) 3
36. 36.What is the smallest number you must add to 68 to make it divisible by 6? (a) 2 (b) 3 (c) 4 (d) 5
37. 37.What is the remainder when 231 is divided by 12? (a) 3 (b) 4 (c) 9 (d) 2
38. 38.What is the remainder when 68 is divided by 3? (a) 0 (b) 1 (c) 2
39. 39.Which of these numbers is divisible by 3? (a) 1849 (b) 1852 (c) 1848 (d) 1847
40. 40.What is the remainder when 198 is divided by 8? (a) 5 (b) 7 (c) 2 (d) 6

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