Let $n$ be an integer such that for any integer $d$, if $d | n$ then $d + 1 | n + 1$.
Show that $n$ is a prime number or equal to $1$
Suppose $n = xy$ for some postive integers $x$ and $y. x|n$, and hence we assume without loss of generality $x + 1|n + 1.$
Then $x|xy + 1$.
However, $xy + x$ is divisible by $x$.
This implies $x − 1$ is divisible by $x$, which is an impossibility if $x$ is not equal to $1$. Therefore whenever $n$ is decomposed into two factors, one must turn out to be $1$. Hence $n$ must be a prime.
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