1. Let and be positive integers and let . Show that the equation has a solution if and only if .
Proof. Suppose has a solution. Then for some . So . From here it follows that and are coprime? Suppose . Then and are coprime. So has no solutions. Hence has a solution.
2. Let be a fixed positive integer greater than . For any integers and , prove that .
Proof. So and where and . Then .
Are these somewhat correct?