Inegalitate cu c.m.m.d.c
Posted: Sun Aug 09, 2009 11:15 pm
Se considera numerele naturale \( x_{ij}\ ,\ i=\overline{1,n}\ ,\ j=\overline{1,m}\ ,\ n\ ,\ m\in\mathbb{N}^{\ast} \) . Sa se arate ca:
\( \prod_{i=1}^n \(x_{i1}\ ,\ x_{i2}\ ,\ \dots\ ,\ x_{im}\)\ \le\ \(\prod_{i=1}^n x_{i1}\ ,\ \prod_{i=1}^n x_{i2}\ ,\ \dots\ ,\ \prod_{i=1}^n x_{im}\) \) .
Am notat \( \(a_1\ ,\ a_2\ ,\ \dots\ ,\ a_m\) \) cel mai mare divizor comun al numerelor naturale \( a_1\ ,\ a_2\ ,\ \dots\ ,\ a_m \) .
\( \prod_{i=1}^n \(x_{i1}\ ,\ x_{i2}\ ,\ \dots\ ,\ x_{im}\)\ \le\ \(\prod_{i=1}^n x_{i1}\ ,\ \prod_{i=1}^n x_{i2}\ ,\ \dots\ ,\ \prod_{i=1}^n x_{im}\) \) .
Am notat \( \(a_1\ ,\ a_2\ ,\ \dots\ ,\ a_m\) \) cel mai mare divizor comun al numerelor naturale \( a_1\ ,\ a_2\ ,\ \dots\ ,\ a_m \) .