Question

Which of the following statement(s) about a perfect square is(are) correct?

• A. It is a product of two identical integers.
• B. The count of distinct prime factors must be divisible by $2.$
• C. None of the above

Answer 1

Answer: (A), (B)

The count of distinct prime factors must be divisible by $2.$

Perfect Square:

• It is the product of two identical integers.
  • For example:

    Perfect Square	Product of Integers
    $4$	$2\times 2$
    $9$	$3\times 3$
    $225$	$15\times 15$

  • As per the above table, a perfect square is the product of two equal integers.
• For any perfect square, the count of distinct prime factors must be divisible by $2.$
  • As a perfect square's prime factors are always found in pairs, the number of different prime factors must be divisible by $2.$
  • For example: $36$ is a perfect square and it's prime factors: $36=2\times 2\times 3\times 3.$ So, $36$ has two prime factors $2$ and $3$, and their counts are $2$ and $2$, respectively, which are divisble by 2.