A camp has a capacity of 700-1100 people. All the people want to participate in a parade. If the parade in rows of 5, 7 or 11 people, then 4, 0 and 7 paople will left respectively. How many people does the camp hold?
We have the congruenes $$x\equiv 4 \pmod 5 \\ x\equiv 0 \mod 7 \\ x\equiv 7\pmod {11}$$ Using the chinese theorem I got $$x=2394 \equiv 84\pmod{385}$$ where $385=5\cdot 7\cdot 11$.
But how does this help us to get the total number of people in the camp?
$385n+84$, where $n$ is an integer needs to be a solution, i.e. between $700$ and $1100$. Just trying the first $3$ possibilities gives $n=2$ as the sole solution, i.e. $854$ people.