What is the greatest common divisor(gcd) of \(2^{1001} - 1\) and \(2^7 - 1\)?
Formula: \(\boxed{ gcd(a^n -1,~a^m-1) = a^{gcd(m,n)}-1 }\)
\(\begin{array}{|rcll|} \hline a &=& 2 \\ n &=& 1001 \\ m &=& 7 \\ \hline gcd(a^n -1,~a^m-1) &=& a^{gcd(m,n)}-1 \\ gcd(2^{1001} - 1,~2^{7} - 1) &=& 2^{gcd(1001,7)}-1 \quad | \quad \mathbf{gcd(1001,7)= 7} \\ gcd(2^{1001} - 1,~2^{7} - 1) &=& 2^{7}-1 \\ \mathbf{gcd(2^{1001} - 1,~2^{7} - 1)} &=& \mathbf{127} \\ \hline \end{array}\)
See Rom: https://web2.0calc.com/questions/what-is-the-greatest-common-divisor-of-2-1001-1-and-2-1012-1