Show that any positive integer N can be written as

Question

Show that any positive integer N can be written as: $N=3^a+2^b*3^c$ or $N=2^b*3^c$, I initially thought it for N = any multiple of 3, but by dividing by 3 it should work for any N

Answer 1

This is false. The number $14$ cannot be written this way.

Clearly $14$ is not of the form $2^a \cdot 3^b$, but it is also straightforward to check that there are no $a,b,c$ such that $14=3^a+2^b\cdot 3^c$, simply check $a,b,c\in\{0,1,2\}$ (numbers get too big outside this range).