Question

The negative of the statement "If a number is divisible by $15$ then it is divisible by $5$ or $3$"

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

Answer 1

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

Let $p,q,r$ be three statements defined as
$p$ : a number $N$ is divisible by $15$
$q$ : number $N$ is divisible by $5$
$r$ : number $N$ is divisible by $3$
Here given statement is $p\to (q\vee r)$
Here negative of above statement is
$\sim (p\to (q\vee r\left)\right)\equiv p\wedge (\sim (q\vee r)$
$\equiv p\wedge (\sim q\wedge \sim r)$
i.e. A number is divisible by $15$ and it is not divisible by $5$ and $3$.