1989 AIME Problems/Problem 3
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Problem
Suppose is a positive integer and
is a single digit in base 10. Find
if
![$\frac{n}{810}=0.d25d25d25\ldots$](http://latex.artofproblemsolving.com/0/d/2/0d2ffb8013a922ca8e69ae1b3c9575aa9b68e7bf.png)
Solution
Repeating decimals represent rational numbers. To figure out which rational number, we sum an infinite geometric series, . Thus
so
. Since 750 and 37 are relatively prime,
must be divisible by 37, and the only digit for which this is possible is
. Thus
and
.
See also
1989 AIME (Problems • Answer Key • Resources) | ||
Preceded by Problem 2 |
Followed by Problem 4 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
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