2005 AIME II Problems/Problem 5
Problem
Determine the number of ordered pairs of integers such that
and
Solution
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Note that the equation can be rewritten as: or
Multiplying through by
and factoring yields
. Therefore,
or
. Therefore, either
or
. For the case
, note that
and
. Thus, all values of a from 2 to 44 will work. For the case
, note that
while
. Therefore, for this case, all values of a from 2 to 12 work. There are
possibilities for the square case and
possibilities for the cube case. Thus, the answer is
.