15. (2005 Croatian National Mathematical Competition) Find all positive integers that can be represented in decimal as $\overline{13 x y 45 z}$, and are divisible by 792, where $x, y, z$ are unknown digits.
15. Solution: Since $792=8 \times 9 \times 11$,

Therefore, the number $\overline{13 x y 45 z}$ is divisible by $8$, $9$, and $11$.
From $8 \mid \overline{13 x y 45 z}$, we know $8 \mid \overline{45 z}$,
and $\overline{45 z}=450+z=448+(z+2)$, thus $8 \mid(z+2)$, so $z=6$.
From $9 \mid \overline{13 x y 45 z}$,
we know $9 \mid(1+3+x+y+4+5+z)$,
and $1+3+x+y+4+5+z=x+y+19=18+(x+y+1)$,
thus, $9 \mid(x+y+1)$,
so $x+y=8$ or 17.
From $1 1 \longdiv { 1 3 x y 4 5 z }$,
we know $11 \mid(6-5+4-y+x-3+1)$,
and $6-5+4-y+x-3+1=x-y+3$,
thus $x-y=-3$ or $x-y=8$.
At this point, there are two possibilities:
(1) If $x+y$ is even, then $x+y=8$ and $x-y=8$, thus $x=8, y=0$;
(2) If $x+y$ is odd, then $x+y=17$ and $x-y=-3$, thus $x=7, y=10$ which is impossible.
Therefore, the only positive integer solution is 1380456.