As N is -1 mod 3, each of N's prime factors must be 1 or -1 mod 3. If they were all 1 mod 3, then their product (which is N) would be 1 mod 3, which is a contradiction. So at least one of them is -1 mod 3.

For the second part, consider the usual proof that there are infinitely many primes and see if you can apply the above result.