Question

How many factors does the number $ 24 $ have?
(A) $ 2 $ factors
(B) $ 4 $ factors
(C) $ 6 $ factors
(D) $ 8 $ factors

Answer 1

Hint: A factor is that number which can completely divide any given number without having any residual value or remainder. Prime factorization is the simplest method for determining the factors of any quantity. The process of writing a number as a result of the product of every prime factor the number is composed of is known as prime factorization.

Complete step by step solution:
We are given a number $ 24 $ , whose number of factors we need to find, so in order to know the number of factors it possesses, we can find out all the different factors.
To pick out each factor of any number, the easiest way is for us to perform a prime factorization for the given number. When we do this, what happens is that we take all the factors of the given number and simplify them until we are able to write each factor as a product of other prime numbers.
The prime numbers we need to remember are all the values lesser than $ 24 $ .
We can list them: $ 2,\;3,\;5,\;7,\;11,\;13,\;17,\;19,\;23 $ .
Now let us write $ 24 $ as the product of any two numbers:
 $ 24 \times 1 = 24 $ , from here we know that ‘ $ 1 $ ’ and ‘ $ 24 $ ’ are factors of $ 24 $ .
Performing prime factorization starting off in this manner \[2\left| \!{\underline {\,
  {24} \,}} \right. \] (this means that we are dividing $ 24 $ by $ 2 $ ):
\[ \Rightarrow 2\left| \!{\underline {\,
  {24} \,}} \right. = 12\]
So the answer we got after the first step of prime factorization is $ 12 $ , so now prime factorize $ 12 $ with $ 2 $ :
\[ \Rightarrow 2\left| \!{\underline {\,
  {12} \,}} \right. = 6\]
Similarly moving forward we get:
\[ \Rightarrow 2\left| \!{\underline {\,
  6 \,}} \right. = 3\]
Now that we have got $ 3 $ (a prime number) after prime factorization of $ 6 $ we can stop the process of prime factorization here.
Express $ 24 $ as the product of every term present:
 $ 24 = 2 \times 2 \times 2 \times 3 $
The combination of each term here yields a factor of $ 24 $ .
The combinations are as follows:
 $ 2,\;4,\;8,\;3,\;6,\;12,\;24 $
Arranging them in ascending order we see:
 $ 2,\;3,\;4,\;6,\;8,\;12,\;24 $
But $ 1 $ is a factor of every number so the factors of $ 24 $ can be listed as:
 $ 1,\;2,\;3,\;4,\;6,\;8,\;12,\;24 $
So there are $ 8 $ factors that make up $ 24 $ .
We have found the count to be a total of $ 8 $ factors that make up the number $ 24 $ . So it is evident that the options: (A) $ 2 $ factors, (B) $ 4 $ factors and (C) $ 6 $ factors are incorrect options. This is because we have found $ 8 $ different factors for the given number and so the other options do not have the correct number of factors given.
Therefore the correct answer is option (D) $ 8 $ factors.

Note: The Greatest Common Factor is the highest number that is a factor of two or more numbers (GCF). The largest number (factor) divides them, yielding a Natural number. Once all of the number's divisors or factors have been identified, there are a handful that are shared by both. The biggest common factor is the highest number found in the common factors.