Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 311 · 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 8^34? (a) 8 (b) 2 (c) 6 (d) 4
2. 2.What is the units digit of 3^23? (a) 3 (b) 9 (c) 1 (d) 7
3. 3.Which of these numbers is divisible by 8? (a) 1760 (b) 1769 (c) 1755 (d) 1764
4. 4.What is the remainder when 762 is divided by 10? (a) 3 (b) 8 (c) 2 (d) 1
5. 5.Which of these numbers is divisible by 11? (a) 2634 (b) 2649 (c) 2654 (d) 2640
6. 6.What is the remainder when 97 is divided by 3? (a) 2 (b) 0 (c) 1
7. 7.What is the smallest number you must add to 886 to make it divisible by 12? (a) 3 (b) 10 (c) 1 (d) 2
8. 8.What is the units digit of 3^16? (a) 1 (b) 3 (c) 9 (d) 7
9. 9.What is the remainder when 94 is divided by 8? (a) 6 (b) 5 (c) 2 (d) 7
10. 10.What is the units digit of 9^14? (a) 9 (b) 0 (c) 1 (d) 2
11. 11.What is the smallest number you must add to 407 to make it divisible by 6? (a) 2 (b) 5 (c) 1 (d) 3
12. 12.What is the units digit of 4^22? (a) 4 (b) 5 (c) 6 (d) 7
13. 13.What is the units digit of 3^35? (a) 7 (b) 3 (c) 9 (d) 1
14. 14.What is the remainder when 806 is divided by 10? (a) 5 (b) 4 (c) 7 (d) 6
15. 15.What is the smallest number you must add to 163 to make it divisible by 3? (a) 2 (b) 1
16. 16.Which of these numbers is divisible by 3? (a) 2557 (b) 2551 (c) 2553 (d) 2554
17. 17.Which of these numbers is divisible by 3? (a) 2692 (b) 2689 (c) 2691 (d) 2695
18. 18.What is the remainder when 265 is divided by 4? (a) 1 (b) 2 (c) 3 (d) 0
19. 19.What is the remainder when 149 is divided by 3? (a) 0 (b) 1 (c) 2
20. 20.What is the remainder when 178 is divided by 11? (a) 9 (b) 1 (c) 3 (d) 2
21. 21.What is the smallest number you must add to 851 to make it divisible by 11? (a) 4 (b) 7 (c) 8 (d) 6
22. 22.Which of these numbers is divisible by 3? (a) 1797 (b) 1802 (c) 1799 (d) 1796
23. 23.What is the remainder when 94 is divided by 6? (a) 2 (b) 3 (c) 4 (d) 5
24. 24.What is the units digit of 7^41? (a) 1 (b) 7 (c) 3 (d) 9
25. 25.What is the units digit of 7^37? (a) 9 (b) 3 (c) 1 (d) 7
26. 26.What is the remainder when 183 is divided by 10? (a) 2 (b) 7 (c) 4 (d) 3
27. 27.What is the units digit of 2^18? (a) 4 (b) 6 (c) 2 (d) 8
28. 28.What is the remainder when 573 is divided by 12? (a) 9 (b) 3 (c) 8 (d) 10
29. 29.What is the smallest number you must add to 229 to make it divisible by 4? (a) 3 (b) 1 (c) 2
30. 30.What is the smallest number you must add to 677 to make it divisible by 11? (a) 6 (b) 5 (c) 7 (d) 4
31. 31.Which of these numbers is divisible by 6? (a) 4968 (b) 4978 (c) 4973 (d) 4963
32. 32.What is the remainder when 117 is divided by 7? (a) 4 (b) 5 (c) 6 (d) 2
33. 33.Which of these numbers is divisible by 11? (a) 7802 (b) 7780 (c) 7795 (d) 7788
34. 34.What is the remainder when 359 is divided by 6? (a) 4 (b) 1 (c) 3 (d) 5
35. 35.What is the units digit of 3^58? (a) 9 (b) 7 (c) 1 (d) 3
36. 36.What is the units digit of 8^44? (a) 8 (b) 6 (c) 4 (d) 2
37. 37.Which of these numbers is divisible by 4? (a) 1895 (b) 1888 (c) 1887 (d) 1890
38. 38.What is the remainder when 161 is divided by 5? (a) 2 (b) 0 (c) 4 (d) 1
39. 39.What is the units digit of 3^43? (a) 9 (b) 7 (c) 3 (d) 1
40. 40.Which of these numbers is divisible by 3? (a) 1534 (b) 1541 (c) 1537 (d) 1536

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