230. Mental Arithmetic. In order to test his pupils' ability in mental arithmetic, Rakebrain one morning asked them to do the following:

- Find two whole numbers (each less than 10), the sum of whose squares plus their product would give a perfect square.

The answer was soon found.
230. There are two solutions not exceeding ten: 3 and 5, 7 and 8.

The general solution is obtained as follows. Denoting the numbers by \(a\) and \(b\), we get

\[
a^{2}+b^{2}+a b=\square=(a-m b)^{2}=a^{2}-2 a m b+b^{2} m^{2}
\]

Therefore,

\[
b+\mathrm{a}=-2 a m+b m^{2}
\]

from which

\[
b=\frac{a(2 m+1)}{m^{2}-1}
\]

where \(m\) can be any integer greater than 1, and \(a\) is chosen so that the number \(b\) is an integer. In general,

\[
\begin{aligned}
& a=m^{2}-1 \\
& 6=2 m+1
\end{aligned}
\]