<s>[INST] $a, b, c, d$ are positive integers， $a-c|a b+c d$, and then $a-c|a d+b c$ 
$a, b, c, d$ are positive integers， $a-c|a b+c d$, and then $a-c|a d+b c$

proof: really easy when use $a b+c d-(a d+b c)$

however my first thought is, $a-c| a b+c d+k(a-c)$, and set some $k$ to prove, failed.
question1 : is this method could be done? any other methods?
question2 : And is there any relations between the question and determinant/matrix
$$\begin{array}{cc} a & d  \\-c & b \end{array}$$
 [/INST] As lab gave you the $k$ which works for question 1, here is the answer to question 2:
$$\det \begin{pmatrix} a & d  \\-c& b \end{pmatrix} = \det \begin{pmatrix} a & b  \\-c & d \end{pmatrix}+\det \begin{pmatrix} a & d-b  \\-c & b-d \end{pmatrix}$$
$$ = \det \begin{pmatrix} a & b  \\-c & d \end{pmatrix}+(b-d)\det \begin{pmatrix} a & -1  \\-c & 1\end{pmatrix} = \det \begin{pmatrix} a & b  \\-c & d \end{pmatrix}+(b-d)\det \begin{pmatrix} a+c & 0  \\-c & 1\end{pmatrix}
$$
The solutions is pretty artificial though...
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