Question 1
For arbitrary positive integers $a, b$, denote $a \ominus b = \frac{a-b}{\text{gcd}(a,b)}$.
Let $n$ be a positive integer. Prove that the following conditions are equivalent:
(i) $\text{gcd}(n, n \ominus m) = 1$ for every positive integer $m \lt n$;
(ii) $n = p^k$ where $p$ is a prime number and $k$ is a non-negative integer.
Let $n$ be a positive integer. Prove that the following conditions are equivalent:
(i) $\text{gcd}(n, n \ominus m) = 1$ for every positive integer $m \lt n$;
(ii) $n = p^k$ where $p$ is a prime number and $k$ is a non-negative integer.