Below we show it's a discrete analog of this: in the plane $\,\mathbb R^2,\,$ a line of negative slope has points in the first quadrant $\,x,y\ge 0\ $ iff its $\,y$-intercept $\,(0,\,y_0)\,$ lies in the first quadrant, i.e. $\, y_0 \ge 0.$
Hint $ $ Normalize $\,k = m\, x + n\, y\,$ so $\,0 \le x < n\,$
by adding a multiple of $\,(-n,m)\,$ to $\,(x,y).$
Lemma $\ \ k = m\ x + n\ y\,$ for some integers $\,x,\,y \ge 0\,$
$\iff$ its normalization has $\,y \ge 0.$
Proof $\ (\Rightarrow)\ $ If $\ x,\, y \ge 0,\,$ normalization adds $\,(-n,m)\,$ zero or more times, preserving $\,y \ge 0.\,$
$(\Leftarrow)\ \,$ If the normalized rep has $\ y < 0,\,$ then $\,k\,$ has no
representation with $\ x,\,y \ge 0,\, $ since to shift so that $\,y > 0\,$
we must add $\,(-n,m)\,$ at least once, which forces $\,x < 0.\ \ $ QED
Finally, notice that since $\ k = m\, x + n\, y\ $ is increasing in both $\,x\,$ and $\,y,\,$ it is clear that the largest non-representable $\,k\,$
has normalization $\,(x,y) = (n\!-\!1,-1),\,$ i.e. the lattice point that is rightmost (max $\,x$) and topmost (max $\,y$) in the nonrepresentable lower half $ (y < 0)$ of the normalized strip, i.e. the vertical strip where $\, 0\le x \le n-1.$ Therefore $\,(x,y) = (n\!-\!1,-1)\,$ yields $\, k = m\,(n\!-\!1)+n\,(-1) = m\,n - m - n\ $ is the largest with no such representation. $\ \ $ QED
Notice that the proof has a vivid geometric picture:
representations of $\,k\,$ correspond to lattice points $\,(x,y)\,$
on the line $\, n\, y + m\, x = k\,$ with negative slope $\, = -m/n.\,$
Normalization is achieved by shifting forward/backward
along the line by integral multiples of the vector $\,(-n,m)\,$
until one lands in the normal strip where $\,0 \le x \le n\!-\!1.\,$ If the normalized rep has $\,y\ge 0\,$ then we are done; otherwise, by the lemma, $\,k\,$ has no rep with both $\,x,y\ge 0.\,$
Remark $ $ There is much literature on this classical problem. To locate such work
you should ensure that you search on the many aliases,
e.g. postage stamp problem, Sylvester/Frobenius problem,
Diophantine problem of Frobenius, Frobenius conductor,
money changing, coin changing, change making problems,
h-basis and asymptotic bases in additive number theory,
integer programming algorithms and Gomory cuts,
knapsack problems and greedy algorithms, etc.