Question

Let \(m_1\), \(m_2\), ..., \(m_k\) be pairwise relatively prime, \(M=m_1m_2\cdots m_k\), \(M_1=M/m_1\), \(M_2=M/m_2\), ..., \(M_k=M/m_k\), and \(M_1y_1\equiv1(\text{mod }m_1)\), \(M_2y_2\equiv1(\text{mod }m_2)\), ..., \(M_ky_k\equiv1(\text{mod }m_k)\). How can I show that\[M_1y_1+M_2y_2+\cdots+M_ky_k\equiv1(\text{mod }M)?\]

Answer 1

@pre-algebra when are you gonna help me with my linear algebra?

Answer 2

If I had to guess, this would be require some application of the Chinese Remainder Theorem.