2003 AIME I Problems/Problem 14
The decimal representation of where and are relatively prime positive integers and contains the digits , and consecutively, and in that order. Find the smallest value of for which this is possible.
To find the smallest value of , we consider when the first three digits after the decimal point are .
Otherwise, suppose the number is in the form of , where is a string of digits and is small as possible. Then . Since is an integer and is a fraction between and , we can rewrite this as , where . Then the fraction suffices.
Thus we have , or
As , we know that the minimum value of is ; hence we need . Since , we need to be divisible by , and this first occurs when (note that if , then ). Indeed, this gives and the fraction ).
|2003 AIME I (Problems • Answer Key • Resources)|
|1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15|
|All AIME Problems and Solutions|
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.