The gcd cannot be a solution as the greatest common divisor, is the largest value, that divides all integers in the set.
But, there can be positive integer values less than gcd, that are not divisor of any of the values in the set.
Say, for the set of numbers:
[11, 22, 33, 44, 55]
have the value of the smallest positive integer, as 6.
The proof for this statement is that for a given set of n numbers:
a_1, a_2, a_3, ..., a_n
considering the integers having only 2 prime factors: let 2, 3; have:
gcd(a_1, a_2, ..., a_n) = min(2^a_1, 2^a_2, ..., 2^{n-1}).min(3^a_1, 3^a_2, ..., 3^{n-1})
The product of the given n numbers is:
2^{a_1+a_2+ ...+ a_n}.3^{a_1+a_2+ ...+ a_n}
So, there can be many positive integers that satisfy the given criteria, of not dividing any of the given numbers.
Such a number can be less than gcd, or greater than the gcd; but don't how to proceed.
Such a number need not to exist.
For example, let $n$ be any integer and $N$ be the set of all positive integers less than $n$.
Obviously $N$ is a finite set.
Now define $m=n\prod\limits_{k\in N}k$.
Then it easy to see that $\gcd(m,n)=n$ and every number smaller than $n$ is divisor of $m$.
Hence for the set $\{m,n\}$ no integer with the properties you want exists.