Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 194 · 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 273 to make it divisible by 6? (a) 5 (b) 2 (c) 3 (d) 4
2. 2.What is the remainder when 79 is divided by 3? (a) 1 (b) 0 (c) 2
3. 3.What is the remainder when 249 is divided by 4? (a) 2 (b) 1 (c) 3 (d) 0
4. 4.What is the remainder when 282 is divided by 9? (a) 4 (b) 3 (c) 2 (d) 6
5. 5.What is the units digit of 8^58? (a) 4 (b) 8 (c) 6 (d) 2
6. 6.What is the units digit of 4^55? (a) 3 (b) 5 (c) 4 (d) 6
7. 7.What is the units digit of 3^16? (a) 7 (b) 9 (c) 3 (d) 1
8. 8.Which of these numbers is divisible by 9? (a) 5700 (b) 5681 (c) 5690 (d) 5688
9. 9.Which of these numbers is divisible by 4? (a) 3400 (b) 3397 (c) 3406 (d) 3401
10. 10.Which of these numbers is divisible by 3? (a) 377 (b) 375 (c) 374 (d) 380
11. 11.What is the smallest number you must add to 505 to make it divisible by 8? (a) 7 (b) 1 (c) 6 (d) 5
12. 12.Which of these numbers is divisible by 11? (a) 3100 (b) 3111 (c) 3102 (d) 3115
13. 13.What is the smallest number you must add to 102 to make it divisible by 7? (a) 4 (b) 3 (c) 2 (d) 5
14. 14.What is the units digit of 3^17? (a) 9 (b) 1 (c) 3 (d) 7
15. 15.What is the smallest number you must add to 546 to make it divisible by 8? (a) 5 (b) 6 (c) 2 (d) 7
16. 16.What is the remainder when 641 is divided by 8? (a) 7 (b) 0 (c) 2 (d) 1
17. 17.Which of these numbers is divisible by 3? (a) 472 (b) 468 (c) 467 (d) 469
18. 18.Which of these numbers is divisible by 11? (a) 5895 (b) 5881 (c) 5874 (d) 5867
19. 19.What is the smallest number you must add to 307 to make it divisible by 5? (a) 4 (b) 1 (c) 2 (d) 3
20. 20.What is the units digit of 3^13? (a) 3 (b) 7 (c) 1 (d) 9
21. 21.What is the remainder when 827 is divided by 11? (a) 3 (b) 1 (c) 2 (d) 9
22. 22.What is the remainder when 136 is divided by 5? (a) 2 (b) 4 (c) 1 (d) 0
23. 23.What is the remainder when 158 is divided by 3? (a) 1 (b) 0 (c) 2
24. 24.What is the units digit of 8^34? (a) 2 (b) 8 (c) 6 (d) 4
25. 25.Which of these numbers is divisible by 3? (a) 2309 (b) 2306 (c) 2307 (d) 2311
26. 26.What is the smallest number you must add to 723 to make it divisible by 11? (a) 3 (b) 4 (c) 2 (d) 8
27. 27.Which of these numbers is divisible by 3? (a) 1420 (b) 1418 (c) 1419 (d) 1423
28. 28.What is the smallest number you must add to 81 to make it divisible by 6? (a) 4 (b) 2 (c) 3 (d) 5
29. 29.What is the remainder when 63 is divided by 4? (a) 1 (b) 0 (c) 2 (d) 3
30. 30.What is the smallest number you must add to 218 to make it divisible by 4? (a) 1 (b) 3 (c) 2
31. 31.What is the remainder when 81 is divided by 4? (a) 0 (b) 1 (c) 2 (d) 3
32. 32.What is the units digit of 3^19? (a) 3 (b) 7 (c) 9 (d) 1
33. 33.What is the units digit of 8^26? (a) 2 (b) 4 (c) 6 (d) 8
34. 34.Which of these numbers is divisible by 4? (a) 1049 (b) 1054 (c) 1048 (d) 1045
35. 35.What is the smallest number you must add to 379 to make it divisible by 6? (a) 4 (b) 3 (c) 5 (d) 1
36. 36.Which of these numbers is divisible by 4? (a) 2252 (b) 2259 (c) 2251 (d) 2255
37. 37.What is the units digit of 4^15? (a) 5 (b) 3 (c) 4 (d) 6
38. 38.What is the units digit of 9^22? (a) 0 (b) 2 (c) 1 (d) 9
39. 39.What is the smallest number you must add to 191 to make it divisible by 10? (a) 8 (b) 9 (c) 1 (d) 7
40. 40.Which of these numbers is divisible by 6? (a) 3502 (b) 3505 (c) 3514 (d) 3504

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