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

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    $\begingroup$ Have you tried induction on $N$? $\endgroup$ – Luke Collins Oct 12 '19 at 20:57
  • $\begingroup$ How can you show it for any multiple of $3$? for example, how can you show it for $3^x\cdot 5^y$? $\endgroup$ – David Diaz Oct 12 '19 at 21:02
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    $\begingroup$ How do you write $14$ for example? $\endgroup$ – user289143 Oct 12 '19 at 21:16

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).


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