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**Janu42** 2. Prove the Chinese remainder theorem: If $\displaystyle m_1,.....,m_n$ are positive integers such that $\displaystyle m_i $ and $\displaystyle m_j$ are relatively prime for i not equal to j, and $\displaystyle k_1,....,k_n$ are any integers, then there is an integer x such that x is congruent to $\displaystyle k_i(mod m_i)$ for 1 less than or equal to i less than or equal to n. (Hint: Consider the generator (1,1,....,1) for $\displaystyle Zm_1$ X......X $\displaystyle Zm_n.$)