Proving 5 integers to be pairwise relatively prime

[DerpyPenguin]

Homework Statement

Let n be an integer. Prove that the integers 6n-1, 6n+1, 6n+2, 6n+3, and 6n+5 are pairwise relatively prime.

Homework Equations

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?

 

[Ray Vickson]

Homework Statement

Let n be an integer. Prove that the integers 6n-1, 6n+1, 6n+2, 6n+3, and 6n+5 are pairwise relatively prime.

Homework Equations

The Attempt at a Solution

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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?

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.