If $a,b,n$ are positive integers such that $\gcd(a+x,b+y) > 1$ for each $x,y\in\{0,1,...,n\}$ then prove that $\min\{a,b\} > n^n$

Question

I am looking for ways to approach the following problem:

If $a,b,n$ are positive integers such that $\gcd(a+x,b+y) > 1$ for each $x,y\in \{0,1,\dots,n\}$ then prove that $\min\{a,b\}\geq n^n$.

But more than that specific thing, I'd like to have a way to accurately enough approximate the smallest the $a$'s and $b$'s have to be in relation to a certain $n$.