Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 416 · 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 635 to make it divisible by 9? (a) 6 (b) 5 (c) 4 (d) 3
2. 2.What is the smallest number you must add to 555 to make it divisible by 12? (a) 3 (b) 10 (c) 8 (d) 9
3. 3.Which of these numbers is divisible by 4? (a) 336 (b) 343 (c) 337 (d) 333
4. 4.What is the smallest number you must add to 310 to make it divisible by 11? (a) 9 (b) 8 (c) 10 (d) 2
5. 5.What is the units digit of 2^36? (a) 8 (b) 4 (c) 6 (d) 2
6. 6.What is the units digit of 4^55? (a) 4 (b) 3 (c) 6 (d) 5
7. 7.What is the remainder when 619 is divided by 10? (a) 1 (b) 8 (c) 7 (d) 9
8. 8.Which of these numbers is divisible by 9? (a) 7490 (b) 7478 (c) 7479 (d) 7484
9. 9.What is the units digit of 3^13? (a) 3 (b) 1 (c) 9 (d) 7
10. 10.What is the remainder when 226 is divided by 7? (a) 1 (b) 5 (c) 2 (d) 3
11. 11.What is the units digit of 8^49? (a) 8 (b) 2 (c) 6 (d) 4
12. 12.What is the units digit of 2^28? (a) 4 (b) 2 (c) 8 (d) 6
13. 13.What is the smallest number you must add to 238 to make it divisible by 3? (a) 1 (b) 2
14. 14.Which of these numbers is divisible by 3? (a) 1340 (b) 1335 (c) 1336 (d) 1334
15. 15.What is the units digit of 7^58? (a) 3 (b) 1 (c) 7 (d) 9
16. 16.What is the units digit of 4^22? (a) 4 (b) 5 (c) 7 (d) 6
17. 17.Which of these numbers is divisible by 9? (a) 4420 (b) 4443 (c) 4430 (d) 4428
18. 18.What is the remainder when 330 is divided by 12? (a) 8 (b) 6 (c) 5 (d) 7
19. 19.Which of these numbers is divisible by 11? (a) 1489 (b) 1512 (c) 1496 (d) 1504
20. 20.What is the remainder when 676 is divided by 10? (a) 4 (b) 6 (c) 7 (d) 5
21. 21.What is the smallest number you must add to 214 to make it divisible by 12? (a) 3 (b) 1 (c) 10 (d) 2
22. 22.Which of these numbers is divisible by 6? (a) 1527 (b) 1540 (c) 1531 (d) 1530
23. 23.What is the smallest number you must add to 414 to make it divisible by 12? (a) 7 (b) 6 (c) 5 (d) 8
24. 24.What is the smallest number you must add to 217 to make it divisible by 5? (a) 3 (b) 1 (c) 2 (d) 4
25. 25.Which of these numbers is divisible by 11? (a) 9490 (b) 9500 (c) 9493 (d) 9514
26. 26.What is the remainder when 464 is divided by 7? (a) 1 (b) 5 (c) 3 (d) 2
27. 27.What is the units digit of 3^45? (a) 7 (b) 1 (c) 9 (d) 3
28. 28.What is the smallest number you must add to 218 to make it divisible by 10? (a) 8 (b) 3 (c) 1 (d) 2
29. 29.What is the smallest number you must add to 327 to make it divisible by 11? (a) 2 (b) 4 (c) 8 (d) 3
30. 30.What is the units digit of 8^27? (a) 2 (b) 6 (c) 4 (d) 8
31. 31.What is the remainder when 258 is divided by 8? (a) 6 (b) 2 (c) 3 (d) 1
32. 32.What is the units digit of 8^54? (a) 4 (b) 2 (c) 8 (d) 6
33. 33.What is the smallest number you must add to 184 to make it divisible by 6? (a) 3 (b) 4 (c) 2 (d) 1
34. 34.Which of these numbers is divisible by 11? (a) 8750 (b) 8775 (c) 8759 (d) 8756
35. 35.Which of these numbers is divisible by 11? (a) 7824 (b) 7832 (c) 7840 (d) 7844
36. 36.What is the smallest number you must add to 564 to make it divisible by 7? (a) 2 (b) 4 (c) 5 (d) 3
37. 37.What is the units digit of 2^16? (a) 2 (b) 4 (c) 8 (d) 6
38. 38.What is the remainder when 203 is divided by 8? (a) 4 (b) 5 (c) 3 (d) 2
39. 39.Which of these numbers is divisible by 4? (a) 2658 (b) 2655 (c) 2661 (d) 2656
40. 40.What is the remainder when 291 is divided by 4? (a) 3 (b) 0 (c) 1 (d) 2

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