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
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 • Resources) | ||
| Preceded by Problem 2 | Followed by Problem 4 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
