Prove that $x$ and $x+1$ are coprime numbers
If $y$ divides $x$ and $x+1$ then it divides $(x+1)-x=1$. Conclude.
$\gcd(x,x+1)=\gcd(x,x+1-x)=\gcd(x,1)=1$.
Hence $x$ and $x+1$ are coprime.
If $x$ is a multiple of $p$, then the next multiple of $p$ is $x+p$, but that's clearly larger than $x+1$.