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In 1997 we proved that if $n$ is of the form $$ 4k, \quad 8k-1\quad {\rm or}\quad 2^{2m+1}(2k-1)+3, $$ where $k,m\in \mathbb N,$ then there are no positiverational numbers $x,y,z$ satisfying $$ xyz = 1, \quad x+y+z = n. $$ Recently,N. X. Tho proved the following statement: let $a\in\mathbb N$ be odd and leteither $n\equiv 0\pmod 4$ or $n\equiv 7\pmod 8$. Then the system of equations$$ xyz = a, \quad x+y+z = an. $$ has no solutions in positive rational numbers$x,y,z.$  A representative example of our result is the following statement: assumethat $a,n\in\mathbb N$ are such that at least one of the following conditionshold:  $\bullet$ $n\equiv 0\pmod 4$  $\bullet$ $n\equiv 7\pmod 8 $  $\bullet$ $a\equiv 0\pmod 4$  $\bullet$ $a\equiv 0\pmod 2$ and $n\equiv 3\pmod 4$  $\bullet$ $a^2n^3=2^{2m+1}(2k-1)+27$ for some $k,m\in \mathbb N.$  Then the system of equations $$ xyz = a, \quad x+y+z = an. $$ has nosolutions in positive rational numbers $x,y,z.$

On integer values of sum and product of three positive rational numbers