Find two such common fractions - one with a denominator of 8, the other with a denominator of 13 - so that they are not equal, but the difference between the larger and the smaller of them is as small as possible.

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Let the first fraction be $x / 8$, and the second be $y / 13$. Then the difference between the larger and the smaller of them is $1 / 104 \mid 13 x-$ $8 y \mid$. The numerator of the resulting fraction is a natural number. Therefore, it cannot be less than $\frac{1}{1} 104$. It can be equal to $1 / 104$. For this to happen, the numerator must be equal to one. This is satisfied, for example, if $x=3, y=5$ (or $x=5, y=8$ ).