number theory

Use Euclidean's Algorithm as follows: 

$11n + 3 \div 6n+7 = 1 \ \text{remainder} \ 5n -4$

$6n + 7 \div 5n - 4 = 1 \ \text{remainder} \ n + 11$

$5n - 4 \div n + 11 = 5 \ \text{remainder} \ -59$

$\gcd(n + 11, 59)$

So, the maximum value for the GCD is $\color{brown}\boxed{59}$, and occurs at $n = 48$