Question

The number of factors of $324$ is ___
A) $15$
B) $12$
C) $9$
D) $21$

Answer 1

Hint: In this question, we need to find the number of factors of $324$. So, we first prime factorize the given number $324$. Then, we express the number $324$ as the product of powers of the constituent prime numbers. Then, we apply the formula for counting the number of factors of a number as $\left( {a + 1} \right)\left( {b + 1} \right)...$ where a, b … and so on are the powers of the constituent prime factors.

Complete step by step solution:
We need to find the prime factors of $324$ using prime factorization.
We know that $324$ is a composite number.
Prime factorization is a method of finding prime numbers which multiply to make the original number.
A prime number is a natural number greater than $1$ that is not a product of two smaller natural numbers.
In prime factorization, we start dividing the number by the first prime number $2$ and continue to divide by $2$ until we get a decimal or remainder. Then divide by $3,5,7,....$ etc. until we get the remainder $1$ with the factors as prime numbers. Then write the numbers as a product of prime numbers.
So, we get,
\[\begin{align}
  & 2\left| \!{\underline {\,
  324 \,}} \right. \\
 & 2\left| \!{\underline {\,
  162\,}} \right. \\
 & 3\left| \!{\underline {\,
  81\,}} \right. \\
 & 3\left| \!{\underline {\,
  27\,}} \right. \\
 & 3\left| \!{\underline {\,
  9\,}} \right. \\
 & 3\left| \!{\underline {\,
  3 \,}} \right. \\
\end{align}\]
Thus, prime factorization of $324$ is,
$324 = 2 \times 2 \times 3 \times 3 \times 3 \times 3$
Expressing as powers of prime factors, we get,
$324 = {2^2} \times {3^4}$
Now, we know the formula for finding the number of factors of a number as $\left( {a + 1} \right)\left( {b + 1} \right)...$ where a, b … and so on are the powers of the constituent prime factors.
Therefore, we have the number of factors of the number $324$ as $\left( {2 + 1} \right)\left( {4 + 1} \right) = 3 \times 5 = 15$.

Note: In this question it is important to note here that a composite number is a positive integer that can be formed by multiplying two smaller positive integers. Equivalently, it is a positive integer that has at least one divisor other than $1$ and itself. However, the prime factorization is also known as prime decomposition. And, the prime factorization of the prime number is the number itself and $1$. Every other number can be broken down into prime number factors, but the prime numbers are the basic building blocks of all the numbers.