Tìm a , b , c ∈ N ∗ sao cho ( a − 1/ b ) ( b − 1 /c ) ( c − 1 /a ) ∈ N ∗ .

Hướng dẫn giải

Ta có: \[\left( {a - \frac{1}{b}} \right)\left( {b - \frac{1}{c}} \right)\left( {c - \frac{1}{a}} \right)\]\[ = \left( {\frac{{ab - 1}}{b}} \right)\left( {\frac{{bc - 1}}{c}} \right)\left( {\frac{{ac - 1}}{a}} \right)\]

\[ = \frac{{\left( {ab - 1} \right)\left( {bc - 1} \right)\left( {ac - 1} \right)}}{{abc}}\]\[ = \frac{{\left( {a{b^2}c - ab - bc + 1} \right)\left( {ac - 1} \right)}}{{abc}}\]

\[ = \frac{{{a^2}{b^2}{c^2} - a{b^2}c - {a^2}bc + ab - ab{c^2} + bc + ac + 1}}{{abc}}\]

\[ = abc - a - b - c + \frac{1}{a} + \frac{1}{b} + \frac{1}{c} + \frac{1}{{abc}}\]

\[ = abc - (a + b + c) + \left( {\frac{1}{a} + \frac{1}{b} + \frac{1}{c}} \right) + \frac{1}{{abc}}\]

Để \(\left( {a - \frac{1}{b}} \right)\left( {b - \frac{1}{c}} \right)\left( {c - \frac{1}{a}} \right) \in \mathbb{N}*\) thì \(a\,,\,\,b\,,\,\,c \in \mathbb{N}*\) và \[abc \in \] Ư\[\left( 1 \right) = \left\{ 1 \right\}.\]

Với \[a = b = c = 1\], ta có: \[abc - (a + b + c) + \left( {\frac{1}{a} + \frac{1}{b} + \frac{1}{c}} \right) + \frac{1}{{abc}}\]

\[ = 1 \cdot 1 \cdot 1 - \left( {1 + 1 + 1} \right) + \left( {1 + 1 + 1} \right) + 1 = 2\].

Vậy \[a = b = c = 1\].