Why is that the smallest positive element in the ideals of the form $a_1\mathbb Z + a_2\mathbb Z+...$ is the greatest common divisor of the coefficients $a_1, a_2...$?
I have seen a proof of that minimum positive element divides all of other elements without remainder but i can't understand how it can be the greatest common divisor. I can understand that any element in the ideal will be a multiple of the common divisors of the coefficients since we can factorize them to be so, but i can't see why it is the GCD. I am looking for intuition.
The GCD of the $a_i$s is a linear combination over $\mathbb{Z}$ of the $a_i$ from euclid GCD algorithm, hence $d\mathbb{Z}\subseteq I$, $I$ being your addition ideal $\sum_i a_i\mathbb{Z}$. Also, as $d\mid a_i\;\forall i$, we get $I\subseteq d\mathbb{Z}$. So, $I=d\mathbb{Z}$.
So, GCD is $d$, and it is the smallest linear combination as $d$ being generator of $I$ has to be the smallest element.