Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 36 · 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 3^40? (a) 9 (b) 1 (c) 7 (d) 3
2. 2.Which of these numbers is divisible by 3? (a) 1673 (b) 1674 (c) 1678 (d) 1675
3. 3.What is the units digit of 2^25? (a) 2 (b) 8 (c) 4 (d) 6
4. 4.What is the remainder when 183 is divided by 7? (a) 2 (b) 1 (c) 6 (d) 0
5. 5.What is the units digit of 8^13? (a) 4 (b) 2 (c) 8 (d) 6
6. 6.What is the smallest number you must add to 179 to make it divisible by 12? (a) 3 (b) 2 (c) 1 (d) 11
7. 7.What is the remainder when 53 is divided by 4? (a) 1 (b) 2 (c) 3 (d) 0
8. 8.What is the units digit of 3^32? (a) 9 (b) 1 (c) 3 (d) 7
9. 9.What is the smallest number you must add to 309 to make it divisible by 5? (a) 1 (b) 3 (c) 4 (d) 2
10. 10.What is the smallest number you must add to 230 to make it divisible by 12? (a) 2 (b) 9 (c) 11 (d) 10
11. 11.What is the remainder when 428 is divided by 8? (a) 4 (b) 3 (c) 6 (d) 5
12. 12.What is the remainder when 194 is divided by 3? (a) 2 (b) 1 (c) 0
13. 13.Which of these numbers is divisible by 11? (a) 6836 (b) 6827 (c) 6831 (d) 6852
14. 14.What is the remainder when 249 is divided by 8? (a) 0 (b) 7 (c) 1 (d) 2
15. 15.What is the units digit of 3^30? (a) 7 (b) 3 (c) 9 (d) 1
16. 16.Which of these numbers is divisible by 4? (a) 2472 (b) 2474 (c) 2478 (d) 2470
17. 17.What is the units digit of 2^43? (a) 2 (b) 8 (c) 6 (d) 4
18. 18.Which of these numbers is divisible by 8? (a) 5859 (b) 5865 (c) 5856 (d) 5851
19. 19.What is the units digit of 8^39? (a) 8 (b) 4 (c) 2 (d) 6
20. 20.What is the units digit of 7^53? (a) 1 (b) 7 (c) 9 (d) 3
21. 21.What is the units digit of 9^40? (a) 9 (b) 0 (c) 2 (d) 1
22. 22.What is the remainder when 74 is divided by 4? (a) 0 (b) 1 (c) 3 (d) 2
23. 23.What is the smallest number you must add to 177 to make it divisible by 5? (a) 4 (b) 3 (c) 1 (d) 2
24. 24.What is the remainder when 91 is divided by 3? (a) 1 (b) 2 (c) 0
25. 25.Which of these numbers is divisible by 9? (a) 6162 (b) 6168 (c) 6165 (d) 6182
26. 26.What is the remainder when 619 is divided by 9? (a) 6 (b) 2 (c) 7 (d) 8
27. 27.What is the smallest number you must add to 55 to make it divisible by 3? (a) 1 (b) 2
28. 28.What is the units digit of 4^44? (a) 4 (b) 7 (c) 5 (d) 6
29. 29.Which of these numbers is divisible by 9? (a) 5344 (b) 5328 (c) 5321 (d) 5330
30. 30.What is the units digit of 3^59? (a) 1 (b) 3 (c) 9 (d) 7
31. 31.Which of these numbers is divisible by 4? (a) 993 (b) 996 (c) 999 (d) 1002
32. 32.What is the remainder when 193 is divided by 4? (a) 0 (b) 3 (c) 1 (d) 2
33. 33.What is the units digit of 2^37? (a) 4 (b) 6 (c) 2 (d) 8
34. 34.What is the units digit of 3^41? (a) 1 (b) 7 (c) 3 (d) 9
35. 35.What is the remainder when 526 is divided by 12? (a) 11 (b) 10 (c) 9 (d) 2
36. 36.What is the remainder when 430 is divided by 7? (a) 4 (b) 3 (c) 5 (d) 2
37. 37.What is the units digit of 2^29? (a) 8 (b) 2 (c) 4 (d) 6
38. 38.What is the smallest number you must add to 574 to make it divisible by 11? (a) 9 (b) 8 (c) 2 (d) 10
39. 39.Which of these numbers is divisible by 3? (a) 1012 (b) 1019 (c) 1014 (d) 1015
40. 40.What is the units digit of 2^10? (a) 6 (b) 2 (c) 4 (d) 8

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