1.Prove that the unmodified claim (retaining the word “other”) remains true if the bounds on k are changed from k ∈ 2, 3, . . . ,(√) n to k ∈ (√ n) , . . . , n − 1 . 2. Prove that the bound cannot be...

1.Prove that the unmodified claim (retaining the word “other”) remains true if the bounds on k are changed from k ∈ 2, 3, . . . ,(√) n to k ∈ (√ n) , . . . , n − 1 .

2. Prove that the bound cannot be changed from k ∈ 2, 3, . . . ,(√ n) to k ∈ (√ n/2) , . . . ,( 3 √ n/2) . That is, prove that the following claim is false: A positive integer n ≥ 2 is evenly divisible by some other integer k ∈ (√ n/2) , . . . ,