Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 797 · 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 remainder when 41 is divided by 4? (a) 3 (b) 0 (c) 2 (d) 1
2. 2.What is the remainder when 42 is divided by 4? (a) 2 (b) 1 (c) 0 (d) 3
3. 3.What is the remainder when 67 is divided by 4? (a) 1 (b) 0 (c) 3 (d) 2
4. 4.Which of these numbers is divisible by 6? (a) 1498 (b) 1490 (c) 1484 (d) 1488
5. 5.What is the smallest number you must add to 139 to make it divisible by 3? (a) 1 (b) 2
6. 6.What is the remainder when 762 is divided by 11? (a) 2 (b) 4 (c) 8 (d) 3
7. 7.What is the remainder when 246 is divided by 4? (a) 0 (b) 3 (c) 2 (d) 1
8. 8.Which of these numbers is divisible by 9? (a) 7871 (b) 7875 (c) 7877 (d) 7889
9. 9.Which of these numbers is divisible by 4? (a) 1332 (b) 1339 (c) 1331 (d) 1335
10. 10.Which of these numbers is divisible by 4? (a) 986 (b) 993 (c) 988 (d) 989
11. 11.Which of these numbers is divisible by 4? (a) 3545 (b) 3543 (c) 3544 (d) 3549
12. 12.Which of these numbers is divisible by 4? (a) 266 (b) 264 (c) 271 (d) 262
13. 13.What is the units digit of 2^20? (a) 8 (b) 2 (c) 4 (d) 6
14. 14.What is the remainder when 474 is divided by 8? (a) 6 (b) 2 (c) 3 (d) 1
15. 15.What is the remainder when 568 is divided by 10? (a) 2 (b) 8 (c) 9 (d) 7
16. 16.What is the units digit of 9^54? (a) 0 (b) 9 (c) 1 (d) 2
17. 17.Which of these numbers is divisible by 6? (a) 2662 (b) 2666 (c) 2675 (d) 2664
18. 18.Which of these numbers is divisible by 11? (a) 6336 (b) 6343 (c) 6353 (d) 6328
19. 19.What is the units digit of 8^21? (a) 6 (b) 2 (c) 4 (d) 8
20. 20.What is the smallest number you must add to 282 to make it divisible by 9? (a) 7 (b) 3 (c) 5 (d) 6
21. 21.What is the remainder when 38 is divided by 3? (a) 1 (b) 0 (c) 2
22. 22.Which of these numbers is divisible by 11? (a) 8484 (b) 8472 (c) 8502 (d) 8481
23. 23.What is the remainder when 389 is divided by 5? (a) 2 (b) 3 (c) 1 (d) 4
24. 24.Which of these numbers is divisible by 9? (a) 2673 (b) 2674 (c) 2669 (d) 2688
25. 25.Which of these numbers is divisible by 9? (a) 7602 (b) 7587 (c) 7581 (d) 7593
26. 26.What is the smallest number you must add to 242 to make it divisible by 5? (a) 1 (b) 4 (c) 3 (d) 2
27. 27.Which of these numbers is divisible by 3? (a) 1228 (b) 1232 (c) 1234 (d) 1230
28. 28.What is the remainder when 707 is divided by 9? (a) 6 (b) 5 (c) 7 (d) 4
29. 29.What is the remainder when 193 is divided by 8? (a) 7 (b) 0 (c) 1 (d) 2
30. 30.What is the units digit of 3^25? (a) 1 (b) 3 (c) 7 (d) 9
31. 31.Which of these numbers is divisible by 11? (a) 7508 (b) 7501 (c) 7521 (d) 7502
32. 32.What is the units digit of 3^42? (a) 7 (b) 9 (c) 1 (d) 3
33. 33.What is the smallest number you must add to 74 to make it divisible by 4? (a) 3 (b) 1 (c) 2
34. 34.Which of these numbers is divisible by 8? (a) 6083 (b) 6088 (c) 6101 (d) 6090
35. 35.What is the smallest number you must add to 84 to make it divisible by 5? (a) 4 (b) 3 (c) 2 (d) 1
36. 36.Which of these numbers is divisible by 6? (a) 3610 (b) 3600 (c) 3603 (d) 3598
37. 37.What is the smallest number you must add to 179 to make it divisible by 9? (a) 2 (b) 8 (c) 1 (d) 3
38. 38.What is the remainder when 61 is divided by 6? (a) 0 (b) 1 (c) 5 (d) 2
39. 39.What is the units digit of 4^56? (a) 4 (b) 7 (c) 6 (d) 5
40. 40.Which of these numbers is divisible by 4? (a) 209 (b) 204 (c) 203 (d) 205

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