Let $n \in \mathbb{N}^{*}$ and $q \geq 3$ a prime,
Suppose that $n$ coprime to $\displaystyle {\small\left(\prod_{\substack{a \leq q \\ \text{a prime}}} {\normalsize a} \right)}$ , after many observations i think that exists $m$ verify $n+m$ coprime to $\displaystyle {\small\left(\prod_{\substack{a \leq q \\ \text{a prime}}} {\normalsize a} \right)}$ and $m < 2q$, i am not sure about this claim and i like to know if there are results about that ?