Variant of System of linear equation...

Question

Using CRT can we solve:

13925 mod x = y
13811 mod x = 2y
13697 mod x = 3y
13583 mod x = 4y

If no, how can this be solved?

Answer 1

$\begin{align}{\bf Hint}\ \bmod x\!:\,\ a&\equiv cy\,\Rightarrow\, ad\equiv cdy\\ b &\equiv dy\,\Rightarrow\ bc\equiv cdy\end{align}$ $\,\Rightarrow\,ad\equiv bc\,\Rightarrow\, x\mid ad-bc$

Applied to OP this yields $\,x\mid 14039 = 101\cdot 139\,$ so $\,x = 101\,$ or $\,139.\,$ Only one choice works.