2005 AMC 12B Problems/Problem 19
Problem
Let and
be two-digit integers such that
is obtained by reversing the digits of
. The integers
and
satisfy
for some positive integer
. What is
?
Solution
let , then
where
and
are nonzero digits.
By difference of squares,
For this product to be a square, the factor of must be repeated in either
or
, and given the constraints it has to be
. The factor of 9 is already a square and can be ignored. Now
must be another square, and since
cannot be
or greater then
must equal
or
. If
then
,
,
, which is not a digit. Hence the only possible value for
is
. Now we have
,
,
, then
,
,
,
, and