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### prove that $2^{35}-1$ is divisible by 31 and 127 [duplicate]

Can you give me a hint to how to approach the problem.How one can show that $2^{35}-1$ is a multiple of 31 and 127?
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### Prove that GCD$(n^a - 1,n^b -1)= n^{GCD(a,b)} -1$ [duplicate]

I used Euclid's theorem $a=bq +r$. $n^a -1 = ((n^b)^q )n^r -1$ I don't know how to move forward.
### Proof that $(2^{17}-1)(2^{71}-1)$ divides $2^{1207}-1$ [duplicate]
$(2^{17}-1)(2^{71}-1)$ divides $2^{1207}-1$. How can we prove it elegantly? Related: given odd $k$, if there a way to predict the $j$ such that $2^j-1$ divides $2^k-1$, short of trying?
### Greatest common divisor of $(2^{21}-1,2^{27}-1)$ [duplicate]
Find $\text{gcd}(2^{21}-1,2^{27}-1).$ My proof: We know that $2^{21}-1=(2^3)^7-1=8^7-1=(8-1)(8^6+\dots+8+1)=7(8^6+\dots+8+1)=7N_1$ and \$2^{27}-1=(2^3)^9-1=8^9-1=(8-1)(8^8+\dots+8+1)=7(8^8+\dots+8+1)=...