Find the least [positive integer](https://artofproblemsolving.com/wiki/index.php/Positive_integer) $n$ for which $\frac{n-13}{5n+6}$ is a non-zero reducible fraction.
$\mathrm{(A)\ } 45 \qquad \mathrm{(B) \ }68 \qquad \mathrm{(C) \  } 155 \qquad \mathrm{(D) \  } 226 \qquad \mathrm{(E) \  }\text84$
For the fraction to be reducible, the greatest common factor of the numerator and the denominator must be greater than $1$. By the [Euclidean algorithm](https://artofproblemsolving.com/wiki/index.php/Euclidean_algorithm),
$\gcd(5n+6, n-13)$
$\gcd(5n+6-5(n-13), n-13)$
$\gcd(71, n-13)$
Since $71$ is prime, $n-13$ must be a multiple of $71$, which first occurs when $n=71+13=84$, $\boxed{\text{(E) 84}}$.