# Proving 5 integers to be pairwise relatively prime

1. Sep 3, 2016

### DerpyPenguin

1. The problem statement, all variables and given/known data
Let n be an integer. Prove that the integers 6n-1, 6n+1, 6n+2, 6n+3, and 6n+5 are pairwise relatively prime.

2. Relevant equations

3. The attempt at a solution
I tried to prove that the first two integers in the list are relatively prime.

(6n-1)-(6n+1)=1 (trying to eliminate the n variable)
6n-1-6n-1=1
-2=1, which is obviously not true.
Not sure where to go from here. Is there another way to prove that two integers are relatively prime?

2. Sep 3, 2016

### Ray Vickson

You can make at least some progress by noting that $O = 6n-1$ is an odd number, and then arguing that for any odd number $O$, the pair $O, O+2$ are relatively prime. That also takes care of the pairs $6n+1, 6n+3$, $6n+3, 6n+5$. You also have that $E = 6n+2$ is an even number, and can argue that $E$ and $E+1$ are relatively prime, so that takes care of $6n+2, 6n+3$. That leaves a few more similar pairs to check.

As for showing relative primeness, you just need to show that any factor of one of the numbers fails to be a factor of the other (except for 1, of course). You may be able to do it by contradiction.