Past exam of the mathematics course of the University of Cambridge / 2021 / ia / Paper 4 / 6E / a / ii / Solution

Because $\gcd (m,n)=1$, choose $u,v\in \mathbb{Z}$ with $um+vn=1$ by Bézout's identity. Thensatisfies $c\equiv a(\mathrm{mod}m)$ and $c\equiv b(\mathrm{mod}n)$. 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 toThe first two give $x\equiv 17(\mathrm{mod}35)$; imposing the last gives