There was a trick to it, of course. Mandating that p be prime was not necessary. All it took was that p be odd. As is the case for all primes other than two!
Henceforth, given p any odd number:
(p-1) and (p+1) are both even.
Furthermore, (p-1) and (p+1) are two consecutive even numbers, meaning that one of the two is a multiple of four.
Hence, of (p-1) and (p+1), one is a multiple of 2 and the other is a multiple of 4.
Hence, (p-1).(p+1) is a multiple of 8.
Additionally, (p-1), p and (p+1) being three consecutive numbers, one of them must be a multiple of 3.
In conclusion, (p-1).p.(p+1) is a multiple of 8*3 = 24.
Kudos to grey_wolf_xvii, krdbuni, thenightwolf, kemonotsukai and janetraeness for figuring it out! And honorable mention to unciaa and timduru for the witty answers. :) And thank you to the others for trying!
I've unscreened the comments now.
thenightwolf also makes a good point that it was not that easy for the non-mathematician, because even though there is no advanced math concept involved, it does require a manner of mathematical thinking. I think he's right. So, uh, apologies for my lack of empathy in this.
The same woofiebutt also offers another problem, a lot more mathy:
Let P be a polynomial of degree n such that P(i) = 1/i for i=1,...,n+1.
And as for the rest, perhaps I'll post later on. We'll see.