FOR NATURAL NUMBERS N prove 1 iff 2 iff 3 iff 1 That is prov

FOR NATURAL NUMBERS N

prove 1 iff 2 iff 3 iff 1. That is, prove the following are all logically equivalent

1) n is divisible by

2) n^2 is divisble by 5

3) n^2 is divisble by 25

Solution :- In problem, statement 1 is not complete.

Let us prove the second and third statement.

Let us assume that n² is divisible by 25. That means 25 divides n^2.

So we can write 25k = n^2 for some integer k .

We can write 25 = 5*5 So 25k = 5*5*k

We can write 5*k = m for some integer m.

So we have 5m = n^2

This implies that n^2 is divisible by 5.