Find $n$ knowing some divisibilities

Question

Question :

Determine the natural numbers $n$, given that $3^n-1$ is divisible by $2^n$, and given that $4^n-1$ is divisible by $3^n$.

My Idea :

We can say that $4^n \equiv 0 ( \mod 4 )$ , which means that $3^n\equiv 0 ( \mod 4 )$, because they are divisible.

Then we can say that $3^n-1 \equiv 2 ( \mod 4 )$ , which means that $2^n \equiv 2 ( \mod 4 )$

Now we can write $2^n = 4M+2$, which means that $n<2 \implies n=1,0$

I'm not sure if my idea is right because this is one of my first times working with $\mod{}$ , so can you tell me :
(1) Whether I'm wrong or not
(2) What is the correct solution?

Hope one of you can help me! Thank you!

Answer 1

If $2^n|(3^n-1)$ then $3^n-1\ge2^n\Rightarrow3^n\ge2^n+1$.

$4^n-1$ can be written as $(2^n-1)(2^n+1)$.

Since $2^n-1$ and $2^n+1$ are coprime, if $3^n|(4^n-1)$ then either $3^n|(2^n-1)$ or $3^n|(2^n+1)$.

In both cases we have that $3^n\le2^n+1$.

Since $3^n\ge2^n+1$ and $3^n\le2^n+1$ we must have that $3^n=2^n+1$.

$3^n=2^n+1\Rightarrow3^n-2^n=2^{n-1}+2^{n-2}\cdot3+\ldots+2\cdot3^{n-2}+3^{n-1}\ge2^{n-1}=1\Rightarrow n\le1$

From this $n$ can only be $0$ or $1$, both of which satisfy the two conditions.

Answer 2

(1) Proof is wrong.

There is almost no connection between Divisibility by $2^n$ & by $3^n$ & by $4$ , hence the Implications used are not valid.

When $x \equiv y \mod z$ , it will not imply that $x$ must be less than $z$ in general.

Data given is about Divisibility by $2^n$ & $3^n$ , not about Divisibility by $4$.

Over-all , there is no valid logical flow between the arguments.

(2) Proof can be like this :

$3^n-1 = c_1 2^n$ [ Divisibility ]
$4^n-1 = c_2 3^n$ [ Divisibility ]

Hence $(4/3)^n-(1/3)^n = c_2$
$(1+1/3)^n-(1/3)^n = c_2$
$(1+1/3)^n-(1/3)^n = c_2$
Let $n=2$ (keeping it small to highlight the Issue)
$1+2/3+1/9-1/9 = c_2$
Hence $c_2$ can not be Integer.
Issue is same when $n$ is larger , the last fractional term in the Power will cancel $(1/3)^n - (1/3)^n$
The Previous fractional term will not cancel , hence $c_2$ can not be Integer.

Hence $n$ must be less than $2$ & we can check whether $0,1$ will work.

We get Same Conclusion when we use the other Data.
Criteria given is redundant.