<s>[INST] 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?
 [/INST] Find the smallest integer, $n$, that leaves
$\text{A remainder of $3$ when divided by $5$.}\tag{1}$
$\text{A remainder of $5$ when divided by $7$.}\tag{2}$
$\text{A remainder of $7$ when divided by $11$.}\tag{3}$
Without using moduli, the Chinese Remainder theorem says that the integers $n$ that meet the above three requirements must be of the form $385A + r$ for a unique $0 \le r < 385$; where $385 = 5 \cdot 7 \cdot 11$.
Looking at the first two conditions, there must exists integers $u$ and $v$ such that
$$\text{$n=5u+3\quad$ and $\quad n = 7v + 5$}$$
So
$$\text{$7n=35u+21\quad$ and $\quad 5n = 35v + 25$}$$
We find (Bezout's Identity) that $-2 \cdot 7n + 3 \cdot 5n = n$. So
\begin{align}
   n &= -2 \cdot 7n + 3 \cdot 5n \\
   n &= -2(35u+21) + 3(35v + 25) \\
   n &= 35(-2u+3v) + 33 \\
    n &= 35w + 33 \\
\end{align}
What have we gained from this? The CRT guarantees that every integer $n$ that satisfies conditions $(1)$ and $(2)$ must be of the form $35w + 33$ for some integer $w$.
Condition $(3)$ tells us that there must exists some integer, $x$, such that
$n = 11x + 7$. So
$$\text{$11n=385w+363\quad$ and $\quad 35n = 385x + 245$}$$
We find that $16\cdot 11n - 5 \cdot 35n = n$ So
\begin{align}
   n &= 16\cdot 11n - 5 \cdot 35n \\
   n &= 16(385w+363) - 5(385x + 245) \\
   n &= 385(16w-5x) + 4583\\
   n &= 385(16w-5x) + 11\cdot 385 + 348\\
   n &= 385(16w-5x+11) + 348\\
    n &= 385y + 348\\
\end{align}
A quick check shows that 348 satisfies conditions $(1), (2)$, and $(3)$.
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