I bask in my own ignorance...

I bask in my own ignorance...  Just ran across a bizarre but easy-to-understand mathematical factoid, which led me to this post that I don't understand at all.

Here's the bizarre factoid: take any prime number other than 2 or 3.  Square it, then subtract one.  The result will be an even multiple of 24.  For example, 1,373 is a prime number; 1,373^2 - 1 = 1,885,128 – which, divided by 24 is 78,547.  I have no idea why this should be so, and it seems mighty odd to me that it is.  What the heck is so special about the number 24?

One slight addendum: the starting number doesn't actually have to be prime, it just has to be relatively prime to 24 (which is another way of saying that it can't be divisible by either 2 or 3).  For example, 25 is not a prime number, but 25^2 - 1 = 24 * 26...