Find the smallest positive integer $x$ which is greater than $1$ and relatively prime to $120$ (recall that relatively prime means that the GCD of $x$ and $120$ is $1$)
We observe that the prime factorization of $120$ is equal to $2^3 \cdot 3 \cdot 5$.  It is a relatively quick matter to test that $2$, $3$, $4$, $5$, and $6$ share a prime factor with $120$, but $\boxed{7}$ does not.