Question 30
a.
Prove that there doesn't exist a sequence of positive integers like $a_1, a_2, a_3, \dots$ such that for all $i, j \in \mathbb{Z}^+$ with $i \lt j$
$$
\text{gcd}(a_i + j, a_j + i) = 1.
$$
b.
Assume that $p$ is an odd prime number. Prove that there exists a sequence of positive integers like $a_1, a_2, a_3, \dots$ such that for all $i, j \in \mathbb{Z}^+$ with $i \lt j$, $\text{gcd}(a_i + j, a_j + i)$ is not divisible by $p$.
Prove that there doesn't exist a sequence of positive integers like $a_1, a_2, a_3, \dots$ such that for all $i, j \in \mathbb{Z}^+$ with $i \lt j$
$$
\text{gcd}(a_i + j, a_j + i) = 1.
$$
b.
Assume that $p$ is an odd prime number. Prove that there exists a sequence of positive integers like $a_1, a_2, a_3, \dots$ such that for all $i, j \in \mathbb{Z}^+$ with $i \lt j$, $\text{gcd}(a_i + j, a_j + i)$ is not divisible by $p$.