2002 AIME II Problems/Problem 8
Problem
Find the least positive integer for which the equation
has no integer solutions for
. (The notation
means the greatest integer less than or equal to
.)
Solution
Note that if , then either
,
or
. Either way, we won't skip any natural numbers.
The smallest such that
is
. (The inequality simplifies to
, which is easy to solve by trial, as the solution is obviously
.)
We can now compute:
From the observation above (and the fact that ) we know that all integers between
and
will be achieved for some values of
. Similarly, for
we obviously have
.
Hence the least positive integer for which the equation
has no integer solutions for
is
.
See also
2002 AIME II (Problems • Answer Key • Resources) | ||
Preceded by Problem 7 |
Followed by Problem 9 | |
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