#1: Let be the smallest integer with no prime divisors. Since is composite, we know that: where . But since was the smallest integer with no prime divisors, then there exists some prime such that . See the contradiction?

#2: Let and . Suppose (so as well). What can you say about then? How would you then bring the prime into this?

#3: This is pretty much the fundamental theorem of arithmetic (except for the uniqueness part). Again, we have another proof by contradiction. Suppose was the smallest number that couldn't be written as a product of primes. It must be composite (otherwise it would be prime!), i.e. . But since was the smallest such number, then and can be written as a product of primes. Can you piece all these together?