= Solution
Because $\gcd(m,n)=1$, choose $u,v\in\mathbb Z$ with $um+vn=1$ by <Bezout identity>[Bézout's identity]. Then
$$
c=avn+bum
$$
satisfies $c\equiv a\pmod m$ and $c\equiv b\pmod n$. Any two simultaneous solutions differ by a multiple of both coprime integers and hence by a multiple of $mn$. This proves the two-modulus <Chinese remainder theorem>.
The three congruences reduce to
$$
x\equiv2\pmod5,
\qquad x\equiv3\pmod7,
\qquad x\equiv1\pmod3.
$$
The first two give $x\equiv17\pmod{35}$; imposing the last gives
$$
\boxed{x\equiv52\pmod{105}}.
$$
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