Good evening,
I have a question concerning the euclidean algorithm.
One knows that for $a_1 , \ldots , a_n \in \mathbb{N} $ and $k\in \mathbb{N} $ there exist some $\lambda_i \in \mathbb{Z}$ such that :
$\gcd(a_1, \ldots, a_n) = \frac{1}{k}\sum_{i=1}^n \lambda_i a_i$
Here is my question: can one find a $m_0 \in \mathbb{N}$ that for every $m \geq m_0$ there are scalars $\mu_i \in \mathbb{N}$ such that:
$\gcd(a_1, \ldots , a_n) = \frac{1}{m}\sum_{i=1}^n \mu_i a_i$
Unfortunately I have only very rudimentary knowledge about number theory ...
With best regards Mat