I'm pretty sure I was told it's impossible to identify multiples of $3$, or powers of $3$ in the p-adic ring $\Bbb Z_2$, and this claim makes total sense to me.
Let $f(x)=3x+2^{\nu_2(x)}$
Is it fair and consistent to define the odd factors of the image of $f$, as the numbers $x$ satisfying $2,3\nmid x$?
If so, let this set be $X$.
Is there any element of $\Bbb Z_2$ which is not in $2^m3^nX:m\in\Bbb N,n\in\Bbb Z$?
I wanted to understand better whether, how and why this approach breaks down.