Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 224 · 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 2^14? (a) 6 (b) 8 (c) 4 (d) 2
2. 2.What is the remainder when 401 is divided by 5? (a) 2 (b) 4 (c) 1 (d) 0
3. 3.What is the units digit of 9^54? (a) 0 (b) 9 (c) 1 (d) 2
4. 4.Which of these numbers is divisible by 4? (a) 2780 (b) 2785 (c) 2777 (d) 2782
5. 5.Which of these numbers is divisible by 4? (a) 2850 (b) 2842 (c) 2844 (d) 2847
6. 6.Which of these numbers is divisible by 3? (a) 841 (b) 839 (c) 844 (d) 840
7. 7.What is the remainder when 419 is divided by 7? (a) 1 (b) 4 (c) 5 (d) 6
8. 8.What is the units digit of 9^15? (a) 8 (b) 9 (c) 7 (d) 1
9. 9.What is the smallest number you must add to 262 to make it divisible by 7? (a) 5 (b) 6 (c) 4 (d) 3
10. 10.What is the smallest number you must add to 239 to make it divisible by 12? (a) 11 (b) 1 (c) 3 (d) 2
11. 11.What is the units digit of 8^25? (a) 6 (b) 8 (c) 2 (d) 4
12. 12.What is the remainder when 815 is divided by 11? (a) 10 (b) 1 (c) 2 (d) 0
13. 13.What is the remainder when 366 is divided by 8? (a) 2 (b) 5 (c) 6 (d) 7
14. 14.What is the remainder when 149 is divided by 7? (a) 5 (b) 2 (c) 3 (d) 1
15. 15.Which of these numbers is divisible by 4? (a) 1708 (b) 1706 (c) 1710 (d) 1715
16. 16.What is the smallest number you must add to 190 to make it divisible by 4? (a) 3 (b) 2 (c) 1
17. 17.What is the remainder when 419 is divided by 11? (a) 10 (b) 0 (c) 1 (d) 2
18. 18.What is the smallest number you must add to 181 to make it divisible by 11? (a) 5 (b) 8 (c) 6 (d) 7
19. 19.Which of these numbers is divisible by 3? (a) 680 (b) 677 (c) 683 (d) 678
20. 20.What is the units digit of 4^21? (a) 5 (b) 4 (c) 3 (d) 6
21. 21.What is the smallest number you must add to 118 to make it divisible by 5? (a) 1 (b) 4 (c) 3 (d) 2
22. 22.Which of these numbers is divisible by 4? (a) 3397 (b) 3392 (c) 3391 (d) 3393
23. 23.What is the remainder when 123 is divided by 7? (a) 3 (b) 6 (c) 4 (d) 5
24. 24.What is the units digit of 8^54? (a) 8 (b) 2 (c) 4 (d) 6
25. 25.What is the units digit of 2^59? (a) 2 (b) 8 (c) 4 (d) 6
26. 26.Which of these numbers is divisible by 4? (a) 2447 (b) 2443 (c) 2444 (d) 2449
27. 27.What is the units digit of 2^39? (a) 4 (b) 6 (c) 2 (d) 8
28. 28.What is the units digit of 2^50? (a) 6 (b) 8 (c) 4 (d) 2
29. 29.What is the units digit of 2^20? (a) 4 (b) 2 (c) 8 (d) 6
30. 30.What is the smallest number you must add to 327 to make it divisible by 11? (a) 3 (b) 2 (c) 8 (d) 4
31. 31.What is the smallest number you must add to 878 to make it divisible by 11? (a) 1 (b) 2 (c) 9 (d) 3
32. 32.What is the units digit of 9^24? (a) 1 (b) 2 (c) 9 (d) 0
33. 33.What is the remainder when 280 is divided by 6? (a) 2 (b) 5 (c) 3 (d) 4
34. 34.Which of these numbers is divisible by 6? (a) 3701 (b) 3711 (c) 3702 (d) 3703
35. 35.What is the remainder when 631 is divided by 9? (a) 2 (b) 8 (c) 0 (d) 1
36. 36.What is the units digit of 7^56? (a) 3 (b) 9 (c) 1 (d) 7
37. 37.What is the units digit of 3^15? (a) 3 (b) 7 (c) 9 (d) 1
38. 38.What is the remainder when 73 is divided by 4? (a) 1 (b) 2 (c) 0 (d) 3
39. 39.Which of these numbers is divisible by 8? (a) 408 (b) 405 (c) 410 (d) 419
40. 40.What is the units digit of 2^19? (a) 4 (b) 8 (c) 2 (d) 6

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