1. For natural numbers $k, l, m$ it holds that

$$
\frac{k+m+k l m}{l m+1}=\frac{2051}{404}
$$

Determine all possible values of the product klm.
SOLUTION. Although the equation in the problem contains three unknowns, we will be able to solve it uniquely thanks to the fact that we are looking for a solution only in the set of natural numbers. We will try to separate the integer part from both fractions (which is easy in the case of a numerical fraction):

$$
\frac{k+m+k l m}{l m+1}=\frac{k(1+l m)+m}{l m+1}=k+\frac{m}{l m+1}, \quad \frac{2051}{404}=5 \frac{31}{404}.
$$

Since $0<m<l m+1$, it follows that

$$
0<\frac{m}{l m+1}<1
$$

and therefore $k=5$. From equation (1), we then get for the fractional parts of both numbers

$$
\frac{m}{l m+1}=\frac{31}{404}
$$

The fraction on the right side of (2) is in its simplest form, and similarly, the fraction on the left side (numbers $m$ and $l m+1$ are clearly coprime). From this equality of fractions, we get the equality of numerators and denominators: $m=31$ and $l m+1=404$, from which we can easily calculate $l=13$. The product $k l m$ can thus take only one value $5 \cdot 13 \cdot 31=2015$.