Chinese remainder theorem problems

Question

Suppose $M$ is a set of non-negative integers such whose greatest common divisor is $d$ and such that $m, n \in M$ implies $m+n \in M$. Show that for some, $ k \geq 0$,

$$ \left\{kd, (k+1)d, (k+2)d,\dots\right\} \subset M $$

Answer 1

Write the elements of $M$ as $d\cdot a_1, d\cdot a_2,...,d\cdot a_n$

Because $m,n\in M$ implies $m+n\in M$, it is enough to prove that the statement of the theorem is true for $d=1$ (which is trivial, if it is true for $a_1,a_2,...,a_n$ and we get any integer greater than $k$, for $d\cdot a_1, d\cdot a_2,...,d\cdot a_n$ we will get any integer divisible by $d$ greater than $d\cdot k$)

So we will prove the statement for $d=1$.

Now this problem is the Frobenius Coin Problem, which can be easily proven using Bezout's lemma