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 \(n\).