623x is congruent to 4(mod 679)?
This equation has no solution since $\displaystyle 623x=4\!\!\!\pmod{679}\Longrightarrow 623x\cdot 4^{-1}=1\!\!\!\pmod{679}$ (why does $\displaystyle 4^{-1}\!\!\!\pmod{679}$ exist?) $\displaystyle \Longrightarrow 623$ is a unit modulo $\displaystyle 679 \Longleftrightarrow 623$ is a unit
in $\displaystyle \mathbb{Z}_{679}$ , which is absurd as $\displaystyle 623\,,\,\,679$ are NOT coprime, as you can easily check.
Tonio