= Solution
The <Chinese remainder theorem> says that for pairwise coprime positive moduli $m_1,\ldots,m_s$, every choice of residues determines exactly one residue class modulo $M=\prod_jm_j$. 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 $2+k\equiv6$, so $k=4+7j$. Consequently \b[all integer solutions] are
$$
\boxed{x=62+105j,\qquad j\in\mathbb Z.}
$$
The moduli are pairwise coprime, so the <Chinese remainder theorem> guarantees that this is the entire solution class.
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