since $792=8\cdot 9\cdot 11$, we find that
$1+3+x+y+4+5+z\equiv 0\pmod 9\iff x+y+z\equiv 5\pmod 9$ $(1)$
$1-3+x-y+4-5+z\equiv 0\pmod{11}\iff x-y+z\equiv 3\pmod{11}$ $(2)$
$400+50+z\equiv 0\pmod 8\iff z\equiv 6\pmod 8$ thus $z=6$
so we get from $(2)$, $x-y\equiv-3\pmod{11}$
case 1: if $x-y=-3$ from $(1)$ we get $2y-3\equiv-1\pmod 9\iff y\equiv 1\pmod 9\iff y=1$
so $x=-2$ which is impossible.
case 2: $x-y=8\Longrightarrow 2y+8\equiv 8\pmod 9\iff y\equiv 0\pmod 9\Longrightarrow y=0$
so $x=8$ and we easily see that $792|1380456$
N.T.TUAN wrote:
Find all possible digits $x, y, z$ such that the number $\overline{13xy45z}$ is divisible
by $792.$
792=8*9*11
$\overline{45z}$ <=> $z=6$
$x+y=8$ or $x+y=18$
$x-y+3=11$ or $x-y+3=0$ <=>
x=8,y=0