Semigroup generated by elements $x_1,...,x_n$ with $\mathrm{gcd}(x_1,...,x_n)=1$

Question

Let $x_1,...,x_n\in\mathbb{N}$ be such that $\mathrm{gcd}(x_1,...,x_n)=1$ and suppose that $S$ is the semigroup generated by them. I would like to show that there always exists $x\in S$ such that every natural number larger than $x$ is also in $S$. Can we construct such $x$ explicitly?