SOLUTIONLet \(a\) be the total number of autographs that Alicia has in her collection; and let the fraction of autographs in box #2 be represented by \(\dfrac{x}{7}\). Clearly, \(a\) and \(x\) must both be positive integers.Then, \(\dfrac{1}{5}a + \dfrac{x}{7}a + 303 = a\)Solve for \(a\) in terms of \(x\): \(35\left( {\dfrac{1}{5}a + \dfrac{x}{7}a + 303} \right) = 35a\;\;\; \Rightarrow \;\;\;7a + 5xa + 10605 = 35a\) \(10605 = 28a - 5xa\;\;\; \Rightarrow \;\;\;10605 = a\left( {28 - 5x} \right)\) Hence, \(a = \dfrac{{10605}}{{28 - 5x}}\)For \(a\) to be a positive integer, the denominator \(28 - 5x\) must be a factor of 10605.