Example 1.40 A centipede with 40 legs and a dragon with 3 heads are in the same cage, with a total of 26 heads and 298 legs. If the centipede has 1 head, then the dragon has $\qquad$ legs. (1988, Shanghai Junior High School Competition)
If the centipede has $x$ individuals, and the dragon has $y$ individuals, and each dragon has $n$ legs, according to the problem, we get
$$
\left\{\begin{array}{l}
x+3 y=26, \\
n y+40 x=298 .
\end{array}\right.
$$

The system of equations has 3 unknowns and 2 equations, with two unknowns in each equation, and $x, y, n$ are all positive integers. For such equations (systems), there is the following definition

Definition If an equation (system) has multiple positive integer solutions (etc.), such an equation (system) is called an indeterminate equation (system).

Generally speaking, simple indeterminate equations are solved under certain conditions, usually seeking their integer solutions.

Below, we will introduce the solution methods for simple indeterminate equations through some examples.
From (1), we get $x=26-3 y$.
Since $0 \leqslant x \leqslant 26$, and $x$ takes integer values, we have $0 \leqslant y \leqslant 8 \frac{2}{3}$, and $y$ takes integer values, so $0 \leqslant y \leqslant 8$.
Substituting (3) into (2), we get
$$
\begin{array}{l}
n y+40(26-3 y)=298, \\
n y=120 y-742 . \\
n=\frac{120 y-742}{y} .
\end{array}
$$
$n$ should be a positive integer, thus, $y$ should be a divisor of $120 y-742$.
When $y=1$, $n=\frac{120-742}{1}$ (discard);
When $y=2$, $n=\frac{120 \times 2-742}{2}$ (discard);
When $y=7$, $n=\frac{120 \times 7-742}{7}=14$;
When $y=8$, $n=\frac{120 \times 8-742}{8}$ (discard).
Substituting $y=7$ into (2), we get $x=5$.
Substituting $\left\{\begin{array}{l}x=5, \\ y=7\end{array}\right.$ into (2), we get $n=14$.
Therefore, the dragon has 14 feet.
In the process of solving this problem, analyzing that $y$ is a divisor of $120 y-742$ is the key, which is the origin of the "divisor analysis method".