Is there a possible explanation in plain English how to use the Chinese Reminder Theorem? For example, if it is the problem of

Find the smallest integer that leave a remainder of 3 when divided by
  5, a remainder of 5 when divided by 7, and a remainder of 7 when
  divided by 11

In some explanation such as in this article it started to use mod, and the whole explanation seemed to very difficult to understand. Is there a more plain English explanation of how this is solved?


Find the smallest integer that leave a remainder of 3 when divided by 5, a remainder of 5 when divided by 7, and a remainder of 7 when divided by 11

We want to find a solution to:
$$
\begin{align}
x&\equiv 3 \pmod 5\\
x&\equiv 5 \pmod 7\\
x&\equiv 7 \pmod {11}
\end{align}
$$
The solution is unique $\pmod {385}$.
We can find a simple formula for this, by first finding the solutions to:
$$
\begin{align}
x&\equiv 1 \pmod 5\\
x&\equiv 0 \pmod 7\\
x&\equiv 0 \pmod {11}
\end{align}
$$
$$
\begin{align}
x&\equiv 0 \pmod 5\\
x&\equiv 1 \pmod 7\\
x&\equiv 0 \pmod {11}
\end{align}
$$
$$
\begin{align}
x&\equiv 0 \pmod 5\\
x&\equiv 0 \pmod 7\\
x&\equiv 1 \pmod {11}
\end{align}
$$
By using the Extended Euclidean Algorithm, we find:
$$
\begin{align}
x\equiv 231\pmod {385}\\
x\equiv 330\pmod {385}\\
x\equiv 210\pmod {385}\\
\end{align}
$$
Hence the solution to:
$$
\begin{align}
x&\equiv a \pmod 5\\
x&\equiv b \pmod 7\\
x&\equiv c \pmod {11}
\end{align}
$$
is:
$$231a+330b+210c\pmod{385}$$
For $a=3,b=5,c=7$, we get $693+1650+1470=3813\equiv 348\pmod{385}$.
