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 ..

This is a solution with divisibility properties only:
$$11\mid 538xy \implies 11\mid y-x+8-3+5 = y-x+10$$
so $$11\mid y-x-1\implies y-x-1=0$$
Also $$3\mid 538xy \implies 3\mid y+x+8+3+5 = y+x+16$$ so $$3\mid  y+x+1 = 2(x+1)\implies x\in\{2,5,8\}$$
Now try for each pair $(x,y)\in \{(2,3),(5,6),(8,9)\}$ if $7\mid 538xy$...
