I thought I proved the following two divisibility statements but later I found out I was wrong. Could someone explain them?
1) For primes $q\geq 2$, $2^q-1$ is divisible by some prime $p$ such that $p\equiv 3(mod\, 4)$
2) If $r$ and $s$ are distinct primes, then $gcd(2^s-1, 2^r-1)=1$