Insufficient information in Number Theory

Remember that $\overline{abc} = 100a + 10b + c$ and $\overline{cba} = 100c + 10b + a$ . Then, $\overline{abc} - \overline{cba} = 100a + 10b + c - 100c - 10b - a$ .

Then, we have $\overline{abc} - \overline{cba} = 100(a-c) + (c-a) = 99(a-c)$ .

Since $\overline{abc} - \overline{cba}$ is equal to another three-digit positive integer, then we have $1<a-c\leq 9$ . If $a-c = 1$ , then $\overline{abc} - \overline{cba}$ isn't equal to a three-digit positive integer.

Note that, $p+r$ is always equal to $9$ .

If $a-c = 2, \rightarrow \overline{pqr} = 198$ .

If $a-c = 3, \rightarrow \overline{pqr} = 297$ .

If $a-c=4, \rightarrow \overline{pqr} = 396$ , etc.

We also see that $q$ is always equal to $9$ , so that we can conclude, $\overline{pqr} + \overline{rqp} = \overline{p9r} + \overline{r9p}$ .

And, since $p+r= 9$ , we have $\overline{pqr} + \overline{rqp} = 1089$ .