10.12 A natural number minus 45 is a perfect square, and this natural number plus 44 is also a perfect square. Try to find this natural number.
(China Beijing Junior High School Grade 3 Mathematics Competition, 1981)
［Solution］Let this natural number be $x$. According to the problem, we have
$$
\left\{\begin{array}{l}
x-45=m^{2} \\
x+44=n^{2} .
\end{array}\right.
$$

where $m, n$ are natural numbers.
Eliminating $x$ from (1) and (2), we get
$$
n^{2}-m^{2}=89,
$$

which is
$$
(n-m)(n+m)=89 .
$$

From (1) and (2), we know that $n>m$, and since 89 is a prime number, it follows from (3) that
$$
\left\{\begin{array}{l}
n-m=1, \\
n+m=89
\end{array}\right.
$$

Solving this, we get
$$
n=45 \text {. }
$$

Thus,
$$
x=45^{2}-44=1981 \text {. }
$$

The natural number sought is 1981.