Let r and s be positive integers such that gcd(r, s) = 1. Use the isomorphism in Example 18.15 to show that form, n ∈ Z, there exists an integer x such that x = m (mod r) and x = n (mod s).

Data from in Example 18.15 

We claim that for integers r ands where gcd(r, s) = 1, the rings Z_rs and Z_r x Z_s are isomorphic. Additively, they are both cyclic abelian groups of order rs with generators 1 and (1, 1) respectively. Thus ∅ : Z_rs → Z_r x Z_s defined by ∅(n · 1) = n · (1, 1) is an additive group isomorphism. To check the multiplicative Condition 2 of Definition 18.9, we use the observation preceding this example for the unity (1, 1) in the ring Z_r x Z_s, and compute. 

(∅(nm)= (nm)· (1, 1) = [n · (1, 1)][m · (1, 1)] = ∅(n)∅(m).