I started reading chapter four of "Elementary Number Theory" by Underwood Dudley and, on page 39, it says
Theorem 2. The Chinese Remainder Theorem. The system of congruences $$x\equiv a_i\pmod{m_i}, \qquad i=1,\,2,\dots,\,k$$ where $\gcd(m_i,m_j)=1$ if $j\neq i$, has a unique solution modulo $(m_1m_2\dots m_k)$.
What does "a solution modulo $(m_1m_2\dots m_k)$" mean ? Don't you have to specify specify what the solution is congruent to ?
I was expecting something like " a unique solution congruent to [something] modulo $(m_1m_2\dots m_k)$.
You are right this is worded badly. The uniqueness bit goes with the "modulo $m_1m_2...m_k$".
So the solution solves $s \equiv a_i \quad mod\ m_i$ for every $i$ and if you have another solution $s'$, then $$ s \equiv s'\ mod\ m_1m_2...m_k $$