We have the product of three consecutive ordinary integers, and when we divide this product by their sum, we get a quotient of 5. $1^{0}$. What are these numbers?

$2^{0}$. In general: in what cases will the quotient obtained in the above manner be an integer? For example, which group of numbers, in terms of magnitude, directly follows the group in $1^{0}$?
$1^{0}$. The three numbers $x-1, x, x+1$; their product

$$
(x-1) x(x+1)=x\left(x^{2}-1\right)
$$

Their sum is $3 x$. The quotient of the product and the sum is

$$
q=\frac{1}{3}\left(x^{2}-1\right)
$$

If $q=5$, then $x^{2}-1=15, x^{2}=16$ and $x=4$ (since we are dealing with a common integer).

The three numbers: $3,4,5$. Their product is 60, their sum is 12 and $\frac{60}{12}=5$.

$2^{0}$. $q$ is an integer if $x-1$ or $x+1$ is a multiple of 3, that is, the smallest or the largest of the three numbers is a multiple of 3. Therefore, such groups of numbers are:

$1,2,3 ; \quad 3,4,5 ; \quad 4,5,6 ; \quad 6,7,8 ; \quad 7,8,9 ; \quad$ etc.

The group consisting of the numbers $3,4,5$ is directly followed by: $4,5,6 .(q=8)$.

Kálmán Matolcsy (Faludy Ferenc g. VI. o. Szombathely.).