Let m and n be positive integers, and let C_m and C_n be the cyclic groups of orders m and n, respectively. 1. Define a

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Let m and n be positive integers, and let C_m and C_n be the
cyclic groups of orders m and n, respectively.
1. Define a group structure on C_m × C_n, the Cartesian product
of C_m and C_n , as
(a, b) · (c, d) = (a · c, b · d)
for a, c ∈ C_m and b, d ∈ C_n . Is C_m × C_n cyclic? If so,
prove it. If not, provide necessary and sufficient conditions on m
and n to guarantee that C_m × C_n is cyclic,
and prove your assertion.
2. Let x ∈ C_35. (Note: x may not be a generator of C_35.)
Suppose you know that exactly two of x^5, x^7, and x^25 are equal.
Determine |x^31|.