# 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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