  Given a prime $p>3$, we characterize positive-definite integral quadratic
forms that are coprime-universal for $p$, i.e. representing all positive
integers coprime to $p$. This generalizes the $290$-Theorem by Bhargava and
Hanke and extends later works by Rouse ($p=2$) and De Benedetto and Rouse
($p=3$). When $p=5,23,29,31$, our results are conditional on the
coprime-universality of specific ternary forms. We prove this assumption under
GRH (for Dirichlet and modular $L$-functions), following a strategy introduced
by Ono and Soundararajan, together with some more elementary techniques
borrowed from Kaplansky and Bhargava. Finally, we discuss briefly the problem
of representing all integers in an arithmetic progression.
