4A. Find all four-digit numbers of the form $\overline{x y z t}$ that satisfy the equations $\overline{z t x}-\overline{x y z}=495$ and $x+y+z=17$.
Solution. From the first equation, we get

$$
x+10 t+100 z-100 x-10 y-z=495
$$

which simplifies to

$$
99(z-x-5)=10(y-t)
$$

The left side of the last equation is divisible by the number 11, which means the right side must also be divisible by 11. The numbers 11 and 10 are coprime, so it must be that $11 \mid y-t$. Since $y, t \in\{0,1, \ldots, 9\}$, it follows that $y=t=0$, or $y=t$. If the right side of (*) is 0, then it must be that $z-x-5=0$, or $z=x+5$. Substituting into the second equation, we get $y=2(6-x)$. Assigning values to the digit $x$, we get $y, z$, and $t$. Finally, the required numbers are: 2878, 4494, and 3686.