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 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