Okay I kind of get your idea here, because p here is prime in the numerator, but nothing in the denominator can divide p because p is a prime, therefore p still exists after the counting and because p still exists therefore it can be divided by p? right
When I was much younger I found a proof of Fermat's Little Theorem using the following observation:
"Every -th number is divisible by ".
Now,
We want to show that,
Is an integer.
Now is not a -th number, nor a -rd number , ... , nor a -th number. Hence one of those numbers: and one of those is a -th thus,
is an integer.