TITLE: Prime factors. Why does $n$ and $n+1$ have no prime factors in common.
QUESTION [7 upvotes]: Could anyone explain to me why $n$ and $n+1$ share no prime factors. I have found some formal proofs online but unfortunately didn't really understand them. 
I accept that it is the case eg: 
$24 = 2\cdot2\cdot2\cdot3$
$25 = 5\cdot5$
$26 = 2\cdot13$
$27 = 3\cdot3\cdot3$
I found this question in a school text book and it was asking to explain why so I assumed there must be an easy explanation (rather than a formal proof).
Thanks, any help would be greatly appreciated.

REPLY [3 votes]: Looking at just the positive integers, what do you think is the smallest prime? If you answer $1$, we have a problem. But if you answer $2$, then we can move forward.
Suppose $n$ is divisible by $2$. What are the two numbers nearest to $n$ that are also divisible by $2$? Obviously $n - 2$ and $n + 2$. Clearly $n - 2 < n - 1 < n < n + 1 < n + 2$.
Now suppose that $n$ is also divisible by some positive odd prime $p$. What are the two numbers nearest to $n$ that are also divisible by $p$? That's right, $n - p$ and $n + p$. Since $p > 2$, we can now say $$n - p < n - 2 < n - 1 < n < n + 1 < n + 2 < n + p.$$