Solve these problems if you're not a retard:

Let $\langle a \rangle = (a_1, a_2, \ldots)$ denote an infinite sequence of positive integers.
$\rightarrow$ Prove that there is no $\langle a \rangle$ such that $\gcd(a_i + j, a_j + i) = 1$ for all $i \neq j$.
Let $p \neq 2$ be a prime.
$\rightarrow$ Prove that there is an $\langle a \rangle$ such that $\gcd(a_i + j, a_j + i) = p$ for all $i \neq j$.