If a natural number $x$ is divisible by $3$

Question

Is the sentence

If a natural number $x$ is divisible by $3$ then, if it is not divisible by $3$ then it is divisible by $5$

true or false?

Answer 1

Consider the statement "a natural number x is divisible by 3" as $p$ and "a natural number x is divisible by 5" as $q$.

Then the given statement can be represented in propositional logic notations as,

$p \Rightarrow (\neg p \Rightarrow q) \\or,p\Rightarrow (\neg(\neg p) \lor q)\\or,p \Rightarrow (p \lor q)\\or, \neg p \lor p \lor q\\or, TRUE $

Thus the statement is always true.

Answer 2

A material conditional is false if and only if it has both a true antecedent and a false consequent. Suppose then that the statement “If a natural number x is divisible by 3 then, if it is not divisible by 3 then it is divisible by 5” is false. This is the case if and only if both "x is divisible by 3" is true and "if x is not divisible by 3 then it is divisible by 5" is false. But the conditional "if x is not divisible by 3 then it is divisible by 5" is false if and only if both "x is not divisible by 3" is true and "x is divisible by 5" is false. So both "x is divisible by 3" is true and "x is not divisible by 3" is true, a contradiction.