SIMON-MARAIS 2019 B2

For each odd prime number $p$ , prove that the integer $$1!+2!+3!+\cdots +p!-\left\lfloor \frac{(p-1)!}{e}\right\rfloor$$ is divisible by $p$ (Here, $e$ denotes the base of the natural logarithm and $\lfloor x\rfloor$ denotes the largest integer that is less than or equal to $x$ .)