Difference between revisions of "2003 AIME I Problems/Problem 14"
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Revision as of 18:58, 4 July 2013
Problem
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.
Solution
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 
, were 
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 
).
See also
| 2003 AIME I (Problems • Answer Key • Resources) | ||
| Preceded by Problem 13 |
Followed by Problem 15 | |
| 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. 
