Question

Show that if n is an integer greater than1, then n can be written as the product of primes

Answer 1

this is called the fundamental theorem of arithmetic
google, you will find several proofs
pick one you understand

Answer 2

Proof by induction: Let P(n) be the proposition that n can be written as the product of primes.

Basis step: P(2) is true, since 2 can be written as the product of one prime, itself.

Induction step: Assume P(j) is true for all j <= k. We need to show that P(k+1) is true.

There are two cases: (a) k+1 is prime. In this case, we immediately conclude that P(k+1) is true.
(b) k+1 is composite. In this case, k+1 = a*b and 2 <= a, b < k+1. By the inductive hypothesis, both a and b can be written as the product of primes. Thus we conclude k+1 is a product of primes.

Because we have completed the basis and inductive steps of mathematical induction, we conclude that P(n) is true for all n >= 2.