Proving the Infinitude of Primes: An Alternative Approach

[k3N70n]

Okay I hope it's okay if I have a couple question. I've been strugelling a bit with this problem set. About a quarter of the questions I just don't seem to see how to start them. Any hints would be greatly appreciated. Thank you kindly
I

Homework Statement

Give that p\nmid n for all primes p\leq \sqrt[3]n show that n> is either prime or the product of two primes.

Homework Equations

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The Attempt at a Solution

I don't really see how to start this one. Any hint would be greatly appreciated

II.

Homework Statement

Give another proof of the infinitude of primes by assuming that there are only finitely many primes say p_1, p_2, ... p_n, and using the following integer to arrive at a a contradiciton:
N = p_2p_3...p_n + p_1p_3...p_n +...+p_1p_2...p_{n-1}

Homework Equations

The Attempt at a Solution

I think that this proof should involve showing that p_k\nmid N\forall k so N must be prime. Which would be like like Euler proof, but I can't seem to see how to set that up

 

[Dick]

For I), if all of the primes less than n^(1/3) don't divide n then n still has a prime factorization. How many factors can there be? For II) for any i, p_i divides all of the terms which sum to N except one. Can it divide N?