exo7 3131

Soient $a_1, \dots, a_n \in \N^{*}$ et $b_i = \prod_{j \ne i} a_j$.

Montrer que :
$\text{pgcd}( a_1, \dots, a_n ) \times \text{ppcm}( b_1, \dots, b_n) = \text{ppcm}( a_1, \dots, a_n ) \times \text{pgcd}( b_1, \dots, b_n) = \prod a_i$.