Largest GCD

If $d$ divides $7n-13$ and $13n+7$ , then it also divides $91n-169$ and $91n+49$ , so it divides their difference, which is $218$ . So $d \le 218$ .

If $n = 33$ , then $7n-13 = 218$ and $13n+7 = 436$ , so the gcd is $218$ . So the maximum is attained: $\fbox{218}$ .

You can use the Euclidean algorithm to solve this problem which states $\gcd (m, n)=\gcd(m-n, n)$ . $\gcd(7n-13, 13n+7)$ $\gcd(7n-13, 6n+20)$ $\gcd(n-33, 6n+20)$ $\gcd(n-33, 218)$

Therefore, the largest value of gcd is $\fbox{218}$