  It is known that for any elliptic curve $E/\mathbb{Q}$ and any integer $m$
co-prime to $30,$ the induced Galois representation $\rho_{E,m}:
\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) \longrightarrow
\text{GL}_{2}(\mathbb{Z}/m\mathbb{Z})$ is surjective if and only if
$\rho_{E,\ell}$ is surjective for any prime $\ell|m.$ In this article, we shall
discuss some generalizations, applications, and variants of this phenomenon.

Number Theory