Problem
NT-B1-M08-P021 CRT for Several Moduli
#21
★★★★☆ Level 4 of 5
Let \(m_1,\ldots,m_s\) be pairwise coprime. Explain why the system \(x\equiv a_i\pmod{m_i}\) has infinitely many integer solutions.
It is enough to use CRT for one solution and then add the product of the moduli.
By the Chinese Remainder Theorem, there is a solution \(x_0\) modulo \(M=m_1m_2\cdots m_s\). Then every number \(x=x_0+kM\), where \(k\in\mathbb Z\), has the same residues modulo all \(m_i\). Since \(k\) can be chosen in infinitely many ways, there are infinitely many solutions.
Theoretical problem on formulating the result.