Too finish off the hint, I'd show that if 6k+1 is divisible by 5, then 6k+5 = 4mod5. Similarily if 6k+5 = 0mod5, then 6k+1 = (-4) = 1mod5. As both of these are clearly not divisible by 2 or 3, it follows that one of them is not divisible by 2,3, or 5.
Show that any set of 8 consecutive integers have the property that at least one member in the set is not divisible by 2,3,5 or 7.
I guess maybe for this one, we would look at things modulo 30, yes?
Show by counterexample that the following statement is false:
Any set of 2k consecutive integers (all greater than p_k) have the property that at least one member of the set is not divisible by the any of the primes up to p_k (inclusive).