Is $\gcd(2^m-1,2^n-1) = 2^{\gcd(m,n)}-1$ related to a Hensel lift?

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Is $\gcd(2^m-1,2^n-1) = 2^{\gcd(m,n)}-1$ related to a Hensel lift?

They look similar in respect of having the same "modulo some power of $p$" component. Are they both examples of a "lift"?