In exercise $1.10$ in this book https://www.math.u-bordeaux.fr/~kbelabas/teach/N1MA9W11/book.pdf it says we can solve
$$(x_1,\dots,x_n)\in\Bbb Z^n\mbox{ such that }\pmatrix{a_{11}&a_{12}&\dots&a_{1n}\\\vdots&\vdots&\ddots&\vdots\\ a_{m1}&a_{m2}&\dots&a_{mn}}\pmatrix{x_{1}\\\vdots\\ x_{n}}=\pmatrix{b_{1}\bmod M_{1}\\\vdots\\b_{m}\bmod M_{m}}$$
Can someone please explain how?
Can we solve if $a_{1i}=a_{2i}=\dots=a_{mi}$ at every $i\in\{1,\dots,n\}$?