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Let $p$ be a prime number, and $n$ a positive integer such that $p^2 mid n$. Show that $p^2 \mid 2^n – 1$.

Let $p$ be a prime number, and $n$ a positive integer such that $p^2 mid n$. Show that $p^2 \mid 2^n – 1$.
My Attempt: Since $p^2 mid n$, we have that $2 mid n$, therefore: $2^n \equiv 1 \pmod {p^2}$. But $p$ is a prime and $2\mid p-1$ so $\gcd(2, p-1) = 1$ and $2\mid p-1$ implies that $2\mid p-1 \implies 2\mid p$. Therefore, $p \mid 2^n – 1$.
Is my reasoning correct?

A:

There’s no need to use the fact that $p$ is prime.
Suppose that $2^n\equiv 1\pmod{p^2}$ for some $n$. Then by induction, we have that $2^n\equiv 1\pmod{p}$. Hence, $p\mid 2^n-1$.

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