Past exam of the mathematics course of the University of Cambridge / 2009 / ii / Paper 1 / 1G / Solution

The Chinese remainder theorem says that for pairwise coprime positive moduli $m_1,\dots ,m_s$, every choice of residues determines exactly one residue class modulo $M={\prod }_j{m}_j$. Equivalently $\mathbb{Z}/M\mathbb{Z}\to {\prod }_j\mathbb{Z}/m_j\mathbb{Z}$ is an isomorphism under reduction.

The first two modular congruences combine to $x=2+15k$. Reducing this expression modulo seven gives $2+k\equiv 6$, so $k=4+7j$. Consequently all integer solutions areThe moduli are pairwise coprime, so the Chinese remainder theorem guarantees that this is the entire solution class.