the primes are odds after 5.
x^2 = (x-1)(x+1) +1
so one less than a prime number will be divisible by (x-1) and (x+1)
the two even numbers above and below the prime number will both be divisible by 2, and one of them must be divisible by 4 as well. Show this will give you the divisibility by 8 that you want.
also, here's a hint on divisibility by 3.
pick any three consecutive integers. how many of them are divisible by 3? how can you show this? how is this useful to what you are trying to prove?