An integer then a decimal?

For any two integers \(x\) and \(y\), the notation \(x.y\) represents a number whose integral part is composed of the digits of \(x\) and its fractional part is composed of the digits of \(y\). For example, if \(x=123\) and \(y=456\) then \(x.y=123.456\). Find the sum of all possible solutions of the equation \(\dfrac{a}{b}=b.a\), where \(a\) and \(b\) are relatively prime positive integers.

  • This is from a reviewer I received. So I take no credit.

  • "The sum of all possible solutions" means that if \((a_1,b_1), (a_2, b_2), \dots, (a_n, b_n)\) are all the possible solutions, then we are asked to find \(\displaystyle \sum_{k=1}^{n} a_k+b_k\).

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