Currently, I read some articles about Many-variable CRT (polyvariable CRT)
But I hardly find a proof for it.
As far as I can find, I see that the theorem is stated like this:
Let $k$ and $n$ be a arbitrary positive integers
and suppose $a_{ij}$ are integers (for $1 \leq i \leq k, 1 \leq j \leq n$). Suppose $m_1, \ldots, m_k $ are pairwise coprime integers and $ b_1, \ldots, b_r $ are arbitrary integers. Then, the $k$ simultaneous congruences
\begin{align*}
a_{11}x_1 + a_{12}x_2 + \ldots + a_{1n}x_n &\equiv b_1 \pmod{m_1}, \\
a_{21}x_1 + a_{22}x_2 + \ldots + a_{2n}x_n &\equiv b_2 \pmod{m_2}, \\
&\vdots \\
a_{k1}x_1 + a_{k2}x_2 + \ldots + a_{kn}x_n &\equiv b_k \pmod{m_k},
\end{align*}
have a solution in integers $x_1, \ldots, x_n $ if and only if, for each $ i \leq k $, the GCD $ m_i $ of $ a_{i1}, a_{i2}, \ldots, a_{in} $, divides $b_i$.
Is there an elementary-math approach?