For how many pairs of positive integers n and m is the statement $mn - 8m + 6n =0$ true? 
Each interior angle of a regular polygon with $n$ sides is $\frac{3}{4}$
  of each interior angle of a second regular polygon with $m$ sides.
How many pairs of positive integers $n$ and $m$ are there for which this
  statement is true?

$\frac{(n-2)*180}{n}$ is the value of one interior angle for a polygon with $n$ sides.
Therefore $\frac{(n-2)*180}{n} =\frac{3(m-2)*180}{4m}$ and $mn - 8m + 6n =0$
For how many pairs of positive integers $n$ and $m$ is the statement $mn - 8m + 6n =0$ true?

$$mn-8m+6n=0$$
$$m(n-8)=-6n$$
$$m=\frac {6n}{8-n}$$
The positive integral solutions are $$(m,n)=(2,2),(6,4),(18,6),(42,7),(10,5) $$
