To simplify the fraction \(\frac{27mn^3}{51m^6n}\), we start by simplifying the coefficients and then the variables.
First, let's simplify the coefficients:
\[ \frac{27}{51} = \frac{27 \div 3}{51 \div 3} = \frac{9}{17} \]
Next, we simplify the variables:
For \(m\): \[ \frac{m}{m^6} = m^{1-6} = m^{-5} \]
For \(n\): \[ \frac{n^3}{n} = n^{3-1} = n^{2} \]
Combining these results, we have:
\[ \frac{27mn^3}{51m^6n} = \frac{9}{17} m^{-5} n^{2} \]
Now we express the fraction as:
\[ \frac{9n^{2}}{17m^{5}} \]
Thus, when the fraction is reduced, the exponent on \(m\) is \(5\).
So, the answer is \(\boxed{5}\).