The Chinese Remainder Theorem states: If \(m_1\) and \(m_2\) are relatively prime, the the system of congruences \(N\equiv a_1\) (\(mod\) \(m_1\)), \(N\equiv a_2\) (\(mod\) \(m_2\)) has a unique solution \(mod\) \(m_1m_2.\)

From this theorem, we can generalize and say that if \(m_1\) and \(m_2\) are relatively prime, then we can allow \(a_1\) and \(a_2\) be any two integers. There will exist an integer \(N\) that satisfies the expressions above.