Find such 5-digit number 84x5y, knowing that 198 divides it

Question

My first step was prime factorization of $198$: $$198=2×3×3×11$$

Since $3$ divides it, $3$ has to divide $(8+4+x+5+y)=17+x+y$

Since $2$ divides it than $2$ also divides $y$

And since $11$ divides this number, $11$ also has to divide $(5y+4x+8)$ At this point I don't know what to do next

Answer 1

We can write $198=2 \cdot 9 \cdot 11$. Now, by divisibility rule of $9$, we have: $$9 \mid (x+y+17) \implies 9 \mid (x+y-1)$$ By divisibility rule of $11$, we have: $$11 \mid (8+x+y-4-5) \implies 11 \mid (x+y-1)$$ This thus gives $99 \mid (x+y-1)$. Clearly $x+y-1 < 99$ which shows that $x+y=1$. Thus, one of them has to be zero and the other has to be one. Since $2$ divides the number, $y$ has to be even.

This shows that the only answer is $x=1$ and $y=0$.

Answer 2

Since $11$ divides the number it divides $8-4+x-5+y=x+y-1$ Better than the test for $3$, use the test for $9$, which says $9$ divides $17+x+y$ or $x+y-1$. Gather it all together

$y$ is even
$11|(x+y-1)$
$9|(x+y-1)$

The last two tell us $x+y-1=0$, so from $y$ being even we have $y=0,x=1$