2006 AIME I Problems/Problem 9
Problem
The sequence is geometric with
and common ratio
where
and
are positive integers. Given that
find the number of possible ordered pairs
Solution
The product of and
is a power of 2. Since both numbers have to be integers, this means that a and r are also powers of 2. Now, let
and
:
For y to be an integer, the numerator must be divisible by 2. This occurs when because
.