All integers of the form \(6n-1\), \(6n+1\), \(6n+3\) and \(6n+5\) are odd, and their absolute differences are 2, 4 or 6.
If an integer \(d\) divides the numbers of the form \(6n-1\), \(6n+1\), \(6n+3\) and \(6n+5\), then it also divides their absolute differences (2, 4 or 6). However, 2, 4 and 6 are even, while the integers are odd, so no such divisor \(d\) other than 1 exists.
As for \(6n+2\), it is relatively prime with \(6n+1\) and \(6n+3\) because they are right next to \(6n+2\).
\(6n+2\) also has a difference of 3 with \(6n-1\) and \(6n+5\). Since the only integer that divides 3 is 3 and 1, and 3 can't divide \(6n+2\), then \(6n+2\) is relatively prime with \(6n-1\) and \(6n+5\).