Question

Prove that - For any integer n>1, n has a prime divisor.

Answer 1

For a formal proof, we use strong induction. Suppose that for all integers kk, with 2≤k<n2≤k<n, the number kk has at least one prime factor. We show that nnhas at least one prime factor.

If nn is prime, there is nothing to prove. If nnis not prime, by definition there exist integers aa and bb, with 2≤a<n2≤a<n and 2≤b<n2≤b<n, such that ab=nab=n.

By the induction assumption, aa has a prime factor pp. But then pp is a prime factor of nn.

Every integer greater than 1 has at least one prime divisor. Proof. (By contradiction) Assume there is someinteger greater than 1 with no prime divisors. Then the set of all suchintegers is non-empty, and thus (by thewell-ordering principle) has a least element; call it n.

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