Chinese Remainder Theorem

Find integers x that leave remainders 2, 3, 2 when divided by 3, 5, 7, respectively. [Sun-Tsu, 100 A.D.]

Theorem 33.27 Let , where n_i are pairwise relatively prime and consider the correspondence

$$a \;\leftrightarrow\; (a_1, a_2, \ldots, a_k),$$

where , and a_i = a mod n_ifor i = 1, $\cdots$,k.