1985 AIME Problems/Problem 7
Problem
Assume that ,
,
, and
are positive integers such that
,
, and
. Determine
.
Solution
It follows from the givens that is a perfect fourth power,
is a perfect fifth power,
is a perfect square and
is a perfect cube. Thus, there exist integers
and
such that
,
,
and
. So
. We can factor the left-hand side of this equation as a difference of two squares,
. 19 is a prime number and
so we must have
and
. Then
and so
,
and
.