Every n∈ℤ+-{1} is a prime number or unique product of prime numbers

Suppose that for all integers k such that 2≤k<n, number k has at least one prime factor.

We have to show that n  has at least one prime factor.

if n is prime case is trivial.

if n is not prime.by definition there exits integers a and b with

2≤a<n and 2≤b<n such that a.b=n

By the induction assumption, a has a prime factor p. But then p is a prime factor ofn