Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 305 · 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 142 to make it divisible by 5? (a) 1 (b) 2 (c) 3 (d) 4
2. 2.What is the smallest number you must add to 270 to make it divisible by 7? (a) 4 (b) 3 (c) 2 (d) 5
3. 3.What is the units digit of 3^38? (a) 9 (b) 3 (c) 1 (d) 7
4. 4.What is the smallest number you must add to 227 to make it divisible by 6? (a) 2 (b) 3 (c) 5 (d) 1
5. 5.What is the remainder when 118 is divided by 4? (a) 2 (b) 0 (c) 3 (d) 1
6. 6.Which of these numbers is divisible by 6? (a) 1544 (b) 1557 (c) 1548 (d) 1550
7. 7.What is the units digit of 4^36? (a) 5 (b) 6 (c) 4 (d) 7
8. 8.What is the units digit of 7^27? (a) 9 (b) 3 (c) 1 (d) 7
9. 9.What is the remainder when 439 is divided by 8? (a) 1 (b) 6 (c) 7 (d) 5
10. 10.What is the remainder when 544 is divided by 10? (a) 5 (b) 4 (c) 3 (d) 6
11. 11.What is the units digit of 4^19? (a) 6 (b) 5 (c) 4 (d) 3
12. 12.What is the remainder when 85 is divided by 3? (a) 2 (b) 0 (c) 1
13. 13.Which of these numbers is divisible by 4? (a) 713 (b) 711 (c) 712 (d) 719
14. 14.What is the remainder when 295 is divided by 12? (a) 5 (b) 6 (c) 8 (d) 7
15. 15.What is the units digit of 4^29? (a) 5 (b) 6 (c) 4 (d) 3
16. 16.Which of these numbers is divisible by 6? (a) 804 (b) 807 (c) 803 (d) 812
17. 17.Which of these numbers is divisible by 3? (a) 766 (b) 761 (c) 762 (d) 763
18. 18.Which of these numbers is divisible by 8? (a) 1915 (b) 1924 (c) 1920 (d) 1932
19. 19.What is the remainder when 95 is divided by 6? (a) 5 (b) 3 (c) 1 (d) 4
20. 20.What is the smallest number you must add to 281 to make it divisible by 12? (a) 5 (b) 7 (c) 6 (d) 8
21. 21.What is the smallest number you must add to 293 to make it divisible by 7? (a) 2 (b) 1 (c) 6 (d) 3
22. 22.What is the smallest number you must add to 232 to make it divisible by 12? (a) 4 (b) 9 (c) 8 (d) 7
23. 23.Which of these numbers is divisible by 11? (a) 7301 (b) 7280 (c) 7282 (d) 7289
24. 24.What is the units digit of 8^10? (a) 4 (b) 8 (c) 6 (d) 2
25. 25.Which of these numbers is divisible by 9? (a) 889 (b) 865 (c) 879 (d) 873
26. 26.Which of these numbers is divisible by 4? (a) 3060 (b) 3057 (c) 3062 (d) 3066
27. 27.What is the units digit of 7^32? (a) 7 (b) 3 (c) 1 (d) 9
28. 28.Which of these numbers is divisible by 11? (a) 9889 (b) 9897 (c) 9880 (d) 9902
29. 29.Which of these numbers is divisible by 3? (a) 483 (b) 488 (c) 484 (d) 482
30. 30.What is the remainder when 294 is divided by 4? (a) 0 (b) 3 (c) 2 (d) 1
31. 31.Which of these numbers is divisible by 11? (a) 2283 (b) 2264 (c) 2274 (d) 2266
32. 32.Which of these numbers is divisible by 11? (a) 3011 (b) 3035 (c) 3014 (d) 3018
33. 33.What is the smallest number you must add to 233 to make it divisible by 3? (a) 2 (b) 1
34. 34.What is the remainder when 464 is divided by 11? (a) 3 (b) 1 (c) 2 (d) 9
35. 35.What is the remainder when 512 is divided by 10? (a) 3 (b) 1 (c) 2 (d) 8
36. 36.What is the smallest number you must add to 401 to make it divisible by 5? (a) 2 (b) 1 (c) 4 (d) 3
37. 37.What is the remainder when 293 is divided by 5? (a) 3 (b) 4 (c) 1 (d) 2
38. 38.Which of these numbers is divisible by 8? (a) 1787 (b) 1792 (c) 1798 (d) 1801
39. 39.What is the remainder when 197 is divided by 6? (a) 5 (b) 1 (c) 3 (d) 4
40. 40.What is the units digit of 9^54? (a) 2 (b) 1 (c) 9 (d) 0

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