[Math] Prime factors. Why does $n$ and $n+1$ have no prime factors in common.

Question

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.

Answer 1

Suppose you have a prime $p$ such that $p$ divides $n$ and $n+1$. Then, $p$ must divide their difference. I.e., $p$ divides $(n+1)-n=1$, which is impossible: no prime divides $1$!