A positive integer n is called a perfect number if it is equal to the sum of all positive integer factors 1 ≤ k 1.Prove that at least one perfect number exists. 2.Prove that, for any prime integer p,...

A positive integer n is called a perfect number if it is equal to the sum of all positive integer factors 1 ≤ k <>

1.Prove that at least one perfect number exists.

2.Prove that, for any prime integer p, the positive integer p 2 is not a perfect number.

3.Let n ≥ 10 be any positive integer. Prove that the set {n, n + 1, . . . , n + 5} contains at most two