Is there any way to show that there exists an integer $a$ such that $(a+1)^n - a^n$ is divisible by $p$ if $p$ = $1$ $\pmod n$ with $n$ and $p$ prime.

Question

Given a prime $n$ and a prime $p$ $=$ $1$ $\pmod n$, is it possible to construct (or even prove) that there exists a number of the form $(a+1)^n - a^n$ with $a$ $>$ $0$ such that it is divisible by $p$?

It is conjectured so, since there is a method for doing this with numbers of the form $(a^n-1)/(a-1)$ such that it is divisible by $p$, and $p$ $=$ $1$ $\pmod n$, with $n$ and $p$ prime:

$p-1$ $|$ $n$

Since $p$ is prime, $b^{p-1}$ $=$ $1$ $\pmod p$, using congruence to the $n^{th}$ power,

$b^{(p-1)/n}$ $=$ $a$ $\pmod p$

$a^n$ $=$ $1$ $\pmod p$

$(a^n-1)/(a-1)$ is divisible by $p$.

Is there a similar procedure for numbers of the form $(a+1)^n - a^n$? Thanks in advance.