Question

How do you find the GCF of \[84\] and \[96\] ?

Answer 1

Hint: To find the GCF of the given numbers in the question, first we have to find the prime factorization of each number. Then by observing the common factors in both the numbers, we can get the GCF of the given number.

Complete step by step answer:
Now, as above discussed first we have to find the prime factorization of each number.
First, we have to find the prime factorization of the first number \[84\].
 $ \begin{align}
  & 2\left| \!{\underline {\,
  84 \,}} \right. \\
 & 2\left| \!{\underline {\,
  42 \,}} \right. \\
 & 7\left| \!{\underline {\,
  21 \,}} \right. \\
 & 3\left| \!{\underline {\,
  3 \,}} \right. \\
 & 1 \\
\end{align} $
So, by doing the prime factorization of \[84\] we get,
Prime factors of \[84\] are \[2\times 2\times 7\times 3\].
Now, we have to find the prime factorization of another number \[96\].
\[\begin{align}
  & 2\left| \!{\underline {\,
  96 \,}} \right. \\
 & 2\left| \!{\underline {\,
  48 \,}} \right. \\
 & 2\left| \!{\underline {\,
  24 \,}} \right. \\
 & 2\left| \!{\underline {\,
  12 \,}} \right. \\
 & 2\left| \!{\underline {\,
  6 \,}} \right. \\
 & 3\left| \!{\underline {\,
  3 \,}} \right. \\
 & 1 \\
\end{align}\]
So, by doing the prime factorization of \[96\] we get,
Prime factors of \[96\] are \[2\times 2\times 2\times 2\times 2\times 3\].
So, by doing the prime factorization of the two numbers given separately we got the prime factors of the two numbers each.
Now, we have been asked to find the GCF of the given two numbers.
To find the GCF of the given two numbers, multiply all the common prime factors of the given two numbers \[84\] and \[96\].
We can clearly observe that \[2\] is common two times and \[3\] is common one time.
Therefore, we get the GCF of the given two numbers \[84\] and \[96\] as the product of the common factors.
Therefore, GCF\[=2\times 2\times 3\].
GCF \[=12\].
Therefore, GCF of the given two numbers \[84\] and \[96\] is \[12\].

Note:
 Students should be well aware of the GCF. Students should be well aware of the process to find the GCF of the given numbers. Students should be careful while doing the prime factorization of the given numbers and taking common prime factors. Students should be very careful while observing the common prime factors.