Let's determine the positive integer solutions of the following equation:

$$
x y z + 2 x + 3 y + 6 z = x y + 2 x z + 3 y z
$$
Let's notice that the first three terms on the left side can be found on the right side either as a $z$-multiple or as a $z$-reciprocal - and these are different terms from each other - so from these term pairs, after reduction to 0, the $(z-1)$ factor can be factored out. If we then subtract 6 from both sides, the entire left side can have the $(z-1)$ factored out:

$$
(z-1)(x y-2 x-3 y+6)=-6
$$

It is easy to further show that the second expression in parentheses can also be factored into a product, so our equation is equivalent to the following:

$$
(x-3)(y-2)(z-1)=-6
$$

Here, all three factors are integers, and since $x, y, z>0$,

$$
x-3 \geq-2, \quad y-2 \geq-1, \quad z-1 \geq 0
$$

The absolute value of the right side can be written as the product of three natural numbers in the following two ways: $6 \cdot 1 \cdot 1$ and $3 \cdot 2 \cdot 1$. Combining these, the possible values for the individual factors are as follows:

$$
\begin{aligned}
& (x-3) \text { can be }: \quad-2, \quad-1, \quad 1, \quad 2, \quad 3, \quad 6 \\
& (y-2) \text { can be }: \quad-1, \quad 1,2,3,6 \\
& (z-1) \text { can be }: \quad 1,2,3,6
\end{aligned}
$$

and from these, we need to select one for each factor such that their product is -6.

In the following table, in the subsequent columns, we first list those selections where (1) one of the consecutive factors is 6, and therefore the product of the other two factors is $-1=(-1) \cdot 1=1 \cdot(-1)$, and then those where one factor is 3 and the product of the remaining factors is $-2=(-2) \cdot 1=(-1) \cdot 2$. We immediately wrote down the values of $x, y, z$ as well, and the number of the desired solutions is 10.

![](https://cdn.mathpix.com/cropped/2024_05_02_492898c1d06d5fb44a16g-1.jpg?height=331&width=1079&top_left_y=1209&top_left_x=154)

Attila Rózsa (Veszprém, Lovassy L. Gymn., IV. grade) Marianna Jegess (Székesfehérvár, Teleki Blanka Gymn., III. grade)