Modulus operation to find unknown If the $5$ digit number $538xy$ is divisible by $3,7$ and $11,$ find $x$ and $y$ .
How to solve this problem with the help of modulus operator ?

I was checking the divisibility for 11, 3:
$5-3+8-x+y = a ⋅ 11$ and $5+3+8+x+y = b⋅3$ and I am getting more unknowns ..

As $53800\equiv208\pmod{231},$
$$10x+y=231z-208$$
Now $99\ge10x+y\ge0$
$$99\ge231z-208\ge0$$
$$\implies 2>307/231\ge z\ge208/231>0$$
$$\implies z=1$$
Consequently $$10x+y=23\iff y=23-10x$$
Can you take it from here?
