In this paper, we present the results related to a problem posed by Andrzej Schinzel. Does the number $N_1(n)$ of integer solutions of the equation

$$ \begin{align*}x_1+x_2+\cdots+x_n=x_1x_2\cdot\ldots\cdot x_n,\,\,x_1\ge x_2\ge\cdots\ge x_n\ge 1\end{align*} $$

$N_a(n)$

$$ \begin{align*}x_1+x_2+\cdots+x_n=ax_1x_2\cdot\ldots\cdot x_n,\,\, x_1\ge x_2\ge\cdots\ge x_n\ge 1.\end{align*} $$

$N_2(n)=1$

$2n-3$

$N_2(n)$

$\{n:N_2(n)\le k,\,n\ge 2\}$

Let $K$ be a number field with a ring of integers ${\mathcal{O}}$. We follow Ferraguti and Micheli [‘On the Mertens–Cèsaro theorem for number fields’, Bull. Aust. Math. Soc.93(2) (2016), 199–210] to define a density for subsets of ${\mathcal{O}}$ and use it to find the density of the set of $j$-wise relatively $r$-prime $m$-tuples of algebraic integers. This provides a generalisation and analogue for several results on natural densities of integers and ideals of algebraic integers.

Let $K$ be a number field with ring of integers ${\mathcal{O}}$. After introducing a suitable notion of density for subsets of ${\mathcal{O}}$, generalising the natural density for subsets of $\mathbb{Z}$, we show that the density of the set of coprime $m$-tuples of algebraic integers is $1/{\it\zeta}_{K}(m)$, where ${\it\zeta}_{K}$ is the Dedekind zeta function of $K$. This generalises a result found independently by Mertens [‘Ueber einige asymptotische Gesetze der Zahlentheorie’, J. reine angew. Math. 77 (1874), 289–338] and Cesàro [‘Question 75 (solution)’, Mathesis 3 (1883), 224–225] concerning the density of coprime pairs of integers in $\mathbb{Z}$.