Întrebare numere a,b,c

Buna ziua
Aflati numerele naturale nenule a,b,c astfel ca:
\[(a+\dfrac{1}{bc})(b+\dfrac{1}{ca})(c+\dfrac{1}{ab})\in N\]
multumesc
na:cred ca in afara de solutia evidenta a=b=c=1 mai exista si altele.

\(\frac{{{\left( abc+1 \right)}^{3}}}{{{\left( abc \right)}^{2}}}\in \mathbb{N}\)
\(\frac{{{\left( abc \right)}^{3}}+3{{\left( abc \right)}^{2}}+3abc+1}{{{\left( abc \right)}^{2}}}\in \mathbb{N}\)
\(abc+3+\frac{3}{abc}+\frac{1}{{{\left( abc \right)}^{2}}}\in \mathbb{N}\)
\(abc/1\)
\(abc=1\)
\(a,b,c\in \mathbb{N}\)
\(a=b=c=1\)