Question

Which of the following statement(s) regarding $324$ is(are) true?

• A. Prime factorization of $324=2\times 2\times 3\times 3\times 3\times 3$
• B. Number of distinct prime factors is $2$
• C. Number of distinct prime factors is $6$
• D. It is a perfect square.
• E. It is not a perfect square.

Answer 1

Answer: (A), (B), (D)

It is a perfect square.

$\bullet$ Prime factorization of $324=2\times 2\times 3\times 3\times 3\times 3$ as depicted in below figure.


$\therefore 324=\left(\underset{–}{2\times 2}\right)\times (\underset{–}{3\times 3}\times \underset{–}{3\times 3})$
And, the distinct prime factors occur in pairs.

$\bullet$ The distinct prime factors of $324$ are $2$ and $3$.
$\therefore$ Number of distinct prime factors of $324$ is $2.$

$\bullet 324=2\times 2\times 3\times 3\times 3\times 3$
$\Rightarrow 324=(2\times 3\times 3)\times (2\times 3\times 3)$
$\Rightarrow 324=18\times 18$
$\Rightarrow 324={18}^2$

$\therefore$ The number $324$ is a perfect square, and it is a square of $18$.