Galois representations and composite moduli

It is known that for any elliptic curve $E\mathrm{/}\mathbb{Q}$ 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$\mathbb{Z}$)$issurjectiveifandonlyif$\rho{E,\ell}$issurjectiveforanyprime$\ell|m.$ In this article, we shall discuss some generalizations, applications, and variants of this phenomenon.