jmc

Notice that the question is essentially asking us for the greatest common divisor of $2002$ and $44444$: any number that can be written in the given form must be divisible by the greatest common divisor of $2002$ and $44444$. Conversely, we can find the values of $m$ and $n$ through repeated applications of the Euclidean algorithm. In particular, $$\begin{array}{rl}& \text{gcd}(2002,44444)\\ & =\text{gcd}(2002,44444-22\cdot 2002)\\ & =\text{gcd}(2002,400)\\ & =\text{gcd}(2002-5\cdot (44444-22\cdot 2002),400)\\ & =\text{gcd}(2,400)\\ & =\boxed{2}.\end{array}$$Notice that $$\begin{array}{rl}& 2002-5\cdot (44444-22\cdot 2002)\\ & =2002-5\cdot 44444+110\cdot 2002\\ & =(111)\cdot 2002+(-5)\cdot 44444\\ & =2,\end{array}$$as desired.