Exponents and numbers

Simple proof 1: $2\equiv 2 (mod 3), 2x2=1(mod 3), 2x2x2\equiv2 (mod 3)$ . As you can see the mods cycle between 1 and 2 and never hit 0, so it will never be divisible by 3.

Simple proof 2: prime factorization of $2^n=2x$ 2 $x$ 2 $x$ 2)... Prime factorization of $3y=3\displaystyle\stackrel{.}{ }A\displaystyle\stackrel{.}{ }B\displaystyle\stackrel{.}{ }C\displaystyle\stackrel{.}{ }D...$ (A,B,C,D... Are y's prime factorization), as you can see, 2^n prime factorization has no three in it, therefore it's not divisible by 3 times anything.