Power of exponents

We can take advantage of the fact that since $2 \equiv -1\pmod 3\\$

$2^{ \text{even number}} \equiv 1\pmod 3\\$

$2^{\text{odd number}} \equiv -1\pmod 3\\$

I have found generalized result for natural number n as :

$Case 1:$ If $n$ is $1$ Then ${ 2 }^{ n\quad }mod\quad 3=3$ if $n$ is $1$

$Case 2:$ If $n$ is even Then ${ 2 }^{ n\quad }mod\quad 3=1$

$Case 3:$ If $n$ is odd then ${ 2 }^{ n\quad }mod\quad 3=2$

So $x=1$ and $y=2$

Therefore $x+y mod 2=1$