Exercise 2. For any pair of positive integers $$m$$ and $$n$$, let $$Z_m + Z_n = \{(a,b) | a \in Z_m, b \in Z_n\}$$. For any pair of elements $$(a,b)$$ and $$(c,d)$$ in $$Z_m + Z_n$$ define $$(a,b) + (c,d) = ((a+c)\; mod\; m, (b+d)\; mod\; n)$$. [For example, in
$$Z_3 + Z_4$$, we have (1,2) + (3,4) = (0,1).] This software checks whether or not $$Z_m + Z_n$$ is cyclic. Run the program for the following choice for $$m$$ and $$n$$: (2,2), (2,3), (2,4), (2,5), (3,4), (3,5), (3,6), (3,7), (3,8), (3,9) and (4,6). On the basis of this output, guess how $$m$$ and $$n$$ must be related for $$Z_m + Z_n$$ to be cyclic.

Please enter $$m$$ and $$n$$.