Question

What is the GCF of \[180\], \[108\] and \[75\]?

Answer 1

Hint: We are given a question asking to write the greatest common factor of the given three numbers. GCF refers to those factors or factors which are common in the numbers under consideration. So, in order to find the GCF of the given numbers, we have to first write the factors of the numbers given to us. Then, we will search for those factors which are common in 180, 108 and 75. Hence, we will have the GCF of the three given numbers.

Complete step-by-step solution:
According to the given question, we are asked to find the value of the GCF of \[180\], \[108\] and \[75\].
GCF or we can say Greatest Common Factor or we even call it as GCD (Greatest Common Divisor), is defined as the greatest factor that exists as the factor common to the numbers taken under consideration.
For example – GCF of 4 and 6 is,
\[4=2\times 2\]
\[6=2\times 3\]
As we can see that in the above factors of 4 and 6, only 2 is common to both the numbers so,
\[GCF(4,6)=2\]
The given numbers we have are,
\[180\], \[108\] and \[75\]
We will first factor these numbers and we get the factors as,
\[180=2\times 2\times 3\times 3\times 5\]
\[108=2\times 2\times 3\times 3\times 3\]
\[75=3\times 5\times 5\]
We now have the factors of the given numbers. We can see that these given numbers have only 3 as common and so we have the answer to the question.
Therefore, \[GCF(180,108,75)=3\]

Note: The GCF or the Greatest Common Factor should not be treated or considered similar to the LCM which is Least Common Multiple. Both these concepts are two different concepts of their own and can be said to be the reverse of each other.
We can also carry out the prime factorization of the given numbers to get the factors and then taking out the factors common to the three given numbers in order to find the GCF.
First, we have 180, we get,
\[\begin{align}
  & 2\left| \!{\underline {\,
  180 \,}} \right. \\
 & 2\left| \!{\underline {\,
  90 \,}} \right. \\
 & 3\left| \!{\underline {\,
  45 \,}} \right. \\
 & 3\left| \!{\underline {\,
  15 \,}} \right. \\
 & 5\left| \!{\underline {\,
  5 \,}} \right. \\
 & 1\left| \!{\underline {\,
  1 \,}} \right. \\
\end{align}\]
\[180={{2}^{2}}\times {{3}^{2}}\times 5\]
Next, we have 108, we get,
\[\begin{align}
  & 2\left| \!{\underline {\,
  108 \,}} \right. \\
 & 2\left| \!{\underline {\,
  54 \,}} \right. \\
 & 3\left| \!{\underline {\,
  27 \,}} \right. \\
 & 3\left| \!{\underline {\,
  9 \,}} \right. \\
 & 3\left| \!{\underline {\,
  3 \,}} \right. \\
 & 1\left| \!{\underline {\,
  1 \,}} \right. \\
\end{align}\]
\[108={{2}^{2}}\times {{3}^{3}}\]
Lastly, we have 75, we get,
\[\begin{align}
  & 3\left| \!{\underline {\,
  75 \,}} \right. \\
 & 5\left| \!{\underline {\,
  25 \,}} \right. \\
 & 5\left| \!{\underline {\,
  5 \,}} \right. \\
 & 1\left| \!{\underline {\,
  1 \,}} \right. \\
\end{align}\]
\[75=3\times {{5}^{2}}\]
From the factors of the given numbers, we will the factors common in the three of them as the GCF.
We get,
\[GCF(180,108,75)=3\]