Question

Let $p$ be a natural number.
Statement $1$: there exists a $p$ for which $p^2-1$ is a prime number
Statement $2$: there exists $p$ for which $p^2-1$ has odd number of factors
Statement $3$: there exists $p$ for which $p(p+1)(p+2)(p+3)+1$ has even number of factors

• A. All three statements are incorrect
• B. Exactly one statement is correct
• C. Exactly two statement are correct
• D. All statements are correct

Answer 1

The correct option is B Exactly one statement is correct

Statement $1$:
$p^2-1=(p+1)(p-1)$ can be a prime number for $p=2$.

Statement $2$:
$p^2-1$ is not a perfect square, so it will have even number of factors.

Statement $3$:
$p(p+1)(p+2)(p+3)+1=(p^2+3p)(p^2+3p+2)+1$
$=(p^2+3p{)}^2+2(p^2+3p)+1$
$=(p^2+3p+1{)}^2$ is a perfect square so it will have odd number of factors