smc

Multiply both sides by $mn$ to clear the denominator. Moving all the terms to the right hand side, the equation becomes $0=mn-4n-2m$. Adding 8 to both sides allows us to factor the equation as follows: $(m-4)(n-2)=8$. Since the problem only wants integer pairs $(m,n)$, the pairs are given by the factors of 8, which are 1,2,4, and 8. This results in four pairs, $(5,10),(6,6),(8,4),$ and $(12,3)$, which is answer $\boxed{4}$.