Show that if $k \in \mathbb{Z}$, then the integers $6k-1$, $6k+1$, $6k+2$, $6k+3$, and $6k+5$ are pairwise relatively prime.
I am still new and uncomfortable with proofs. Any help would be great.
[Math] Pairwise relatively prime proof (number theory)
number theory
Best Answer
When you're not sure what to do, look at examples. (This can be a good idea even if you are sure what to do!) In this case,
$$\begin{align} k=1&\Rightarrow 5,7,8,9,11\\ k=2&\Rightarrow 11,13,14,15,17\\ k=3&\Rightarrow 17,19,20,21,23\\ k=4&\Rightarrow 23,25,26,27,29\\ k=5&\Rightarrow 29,31,32,33,35\\ &\vdots \end{align}$$
You might notice that only the middle number in each list is even, and only the one next to it is divisible by $3$. Is it obvious that this should be the case? You might also notice that there's always a multiple of $5$, which moves around. Can you see why that happens? Finally, you might notice that the difference between the largest and smallest numbers in each list is always $6$. So how large can a number $d$ possibly be if it divides two numbers in one of the lists?