Time and Work problem 3 by Dhaval Furia

Let a man does the work in $m$ days. Since a machine does the work in $13\times 2=26$ days, we have

$78+8m=2(208+3m)\implies m=169$ .

So a man does the work in $169$ days and hence to finish the work in $13$ days, $\dfrac{169}{13}$ or $\boxed {13}$ men are required.