1. The lieutenant is engaged in drill training with the new recruits. Upon arriving at the parade ground, he saw that all the recruits were lined up in several rows, with the number of soldiers in each row being the same and 5 more than the number of rows. After the training session, the lieutenant decided to line up the recruits again but couldn't remember how many rows there were. So he ordered them to line up with as many rows as his age. It turned out that the number of soldiers in each row was again equal, but there were 4 more soldiers in each row than in the initial formation. How old is the lieutenant?
Solution. Let $n$ be the number of rows in the original formation. Then, there were originally $n+5$ soldiers in each row, and in the second formation, there were $n+9$ soldiers in each row. Let the age of the lieutenant be $x$. Then, according to the problem, we get the equation

$$
x=\frac{n(n+5)}{n+9} \Rightarrow x=n-4+\frac{36}{n+9}
$$

The number $n+9$ must be a divisor of the number 36. Considering that $n$ and $x$ are natural numbers, we get the following solutions

$$
\left\{\begin{array}{l}
n+9=12 \\
x=2
\end{array},\left\{\begin{array}{l}
n+9=18 \\
x=7
\end{array},\left\{\begin{array}{l}
n+9=36 \\
x=24
\end{array}\right.\right.\right.
$$

From a common sense perspective, the last answer fits, although the other two answers were also considered correct.

Answer: 24 years