Find greater than zero integers for $N=mx+ny$

Question

I would be grateful for some tips on how to answer:

Given integers $m,n>0$ with $gcd(m,n)=1$, show that all integers $N\ge mn$ are expressible as $N=mx+ny $. For $x,y\ge 0 $, where $x,y$ are integers.

I know that as the $gcd=1$ we can find integers $x,y$ s.t: $1=mx+ny$ and therefore $mn=m^2nx+mn^2y$. I am unsure of how to use this to show we can find greater than zero integers.

Answer 1

In the set $\{N, N-m, N-2m,\ldots, N-(n-1)m\}$ all elements are non-negative and distinct modulo $n$. So you can find $0\leq k <n$ such that $N-km= 0\pmod n$. $\Rightarrow\ N= km +k_1n$.