Problem
NT-B2-M09-P018 Avoiding a Finite Set of Residues
#18
★★★★★ Level 5 of 5
Let \(m_1,\ldots,m_k\) be pairwise coprime moduli, and for each \(i\) let one residue \(a_i\pmod {m_i}\) be forbidden. Prove that there are infinitely many integers \(x\) that are not congruent to \(a_i\) modulo \(m_i\) for any \(i\).
For each modulus choose an allowed residue \(b_i\ne a_i\).
Since \(m_i>1\), choose a residue \(b_i\) different from \(a_i\) modulo \(m_i\). By CRT there exists \(x_0\) such that \(x_0\equiv b_i\pmod {m_i}\) for all \(i\). Then all \(x=x_0+tM\), where \(M=m_1\cdots m_k\), have the same residues modulo all \(m_i\). Hence they avoid all forbidden classes, and there are infinitely many such integers.
A good problem showing CRT as a choice tool, not only a divisibility tool.