Question

The negation of the proposition: "If a number is divisible by $15$, then it is divisible by $5$ or $3$" is

• A. If a number is divisible by $15$ then it is not divisible by $5$ and $3$
• B. A number is divisible by $15$ and it is not divisible by $5$ and $3$.
• C. A number is not divisible by $15$ or it is divisible by $5$ or $3$.
• D. A number is not divisible by $15$ or it is not divisible by $5$ and $3$

Answer 1

The correct option is B A number is divisible by $15$ and it is not divisible by $5$ and $3$.

Let p: A number is divisible by $15$
q: It is divisible by $5$ or $3$
We know that $\sim (p\to q)$ is $p\wedge (\sim q)$
So, we have negation of the proposition: "If a number is divisible by $15$, then it is divisible by $5$ or $3$" as "A number is divisible by $15$ and it is not divisible by $5$ and $3$."