Let $m\in\mathbb N$ be any natural number and $f(X)=aX^2+bX+c$ be a polynomial with coefficients $a,b,c\in\mathbb Z$ such that $gcd(a,b,c)=1$.
Is there an $R\in\mathbb Z$ so that $f(R)$ is prime to $m$?
Intuitively, the answer clearly seems yes. But I'm having a hard time proving it. Sure, if $gcd(c,m)=1$, we can simply put $R=0$. My idea would be looking into any prime factor $p^\alpha$ of $m$ and somehow produce simultaneous congruences such that $f(R)\equiv 1 \mod p^\alpha$. But that didn't work out.