Answer
Statement: $\forall$ integers n, if n is divisible by 6, then n is divisible by 2 and n is divisible by 3.
Contrapositive: $\forall$ integers n, if n is not divisible by 2 or n is not divisible by 3, then n is not divisible by 6.
Converse: $\forall$ integers n, if n is divisible by 2 and n is divisible by 3, then n is divisible by 6.
Inverse: $\forall$ integers n, if n is not divisible by 6, the n is not divisible by 2 or n is not divisible by 3.
All the statements are true. The statement and its contrapositive are logically equivalent (the statement is true, therefore its contrapositive is as well). The converse and inverse are contrapositives of each other and are hence logically equivalent. The converse is true, therefore the inverse is true as well.
Work Step by Step
A statement of the form: $\forall x \in D$, if P(x) then Q(x),
has as its contrapositive statement: $\forall x \in D$, if ~Q(x) then ~P(x),
as its converse statement: $\forall x \in D$, if Q(x) then P(x),
and as its inverse statement: $\forall x \in D$, if ~P(x) then ~Q(x).
