2003 AMC 10A Problems/Problem 20
Problem 20
A base-10 three digit number is selected at random. Which of the following is closest to the probability that the base-9 representation and the base-11 representation of
are both three-digit numerals?
Solution
The smallest base-11 number that has 3 digits in base-10 is which is
.
The largest number in base-9 that has 3 digits in base-10 is
Alternatively, you can do 9^3-1
The smallest number in base-9 that has 3 digits in base-10 is
Hence, all numbers that will have 3 digits in base-9, 10, and 11 will be between and
, thus the total amount of numbers that will have 3 digits in base-9, 10, and 11 is
There are 900 possible 3 digit numbers in base 10.
Hence, the answer is
See Also
2003 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 19 |
Followed by Problem 21 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
All AMC 10 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.