Chapter 4.4, Problem 30E

Complete the proof of the Chinese remainder theorem by showing that the simultaneous solution of a system of linear congruences modulo pairwise relatively prime moduli is unique modulo the product of these moduli. [Hint: Assume that x and y are two simultaneous solutions. Show that $m_j|x-y$ for all i. Using Exercise 29, conclude that $m=m_{\text{1}}{m}_{\text{2}}\mathrm{...}{m}_n|x-y$ ].