Answer: E

Small Hint:

Any common divisor of $n-13$ and $5n+6$ also divides a suitable linear combination

Big Hint:

Compute $(5n+6)-5(n-13)$

Solution:

Euclid’s algorithm gives $$\begin{aligned} &\gcd(n-13,5n+6)\\ &\quad=\gcd(n-13,71). \end{aligned}$$ Since $71$ is prime, the nonzero fraction is reducible exactly when $n-13$ is a nonzero multiple of $71.$ The least positive possibility is $n=13+71=84,$ which is not among the four numerical choices.

Thus the correct answer is E.