Question

Prove that every two consecutive integers are co-prime.

Answer 1

Let $n$ and $n+1$ be two consecutive integers.
Let $(n,n+1)=d$ $[$H.C.F. of $n$ and $n+1$ is $d]$
$\therefore d|n\text{and}d|n+1$
$⟹d|(n+1)-n$

$⟹d|1$
$\therefore d=1$
$\therefore (n,n+1)=1$
i.e., $n$ and $(n+1)$ are relatively prime