Question

What is the greatest common factor of $60$ and $72$?

Answer 1

Hint: To obtain the greatest common factor of the given two numbers $60$ and $72$, we need to consider their respective prime factorization. We can use the method of long division for the prime factorization of both the numbers. Then we have to consider all the prime factors common to both the numbers, which are to be multiplied together so as to obtain the final required greatest common factor of the two numbers.

Complete step by step answer:
The two numbers given in the above question are $60$ and $72$. According to the question, we have to find out the greatest common factor of these two numbers. As the term “greatest common factor” suggests, we have to consider all the prime factors which are common for the two numbers. Then to get the greatest common factor, we have to multiply all of the common prime factors. So we consider the prime factorization of $60$ as shown below.
\[\begin{align}
  & 2\left| \!{\underline {\,
  60 \,}} \right. \\
 & 2\left| \!{\underline {\,
  30 \,}} \right. \\
 & 3\left| \!{\underline {\,
  15 \,}} \right. \\
 & 5\left| \!{\underline {\,
  5 \,}} \right. \\
 & \left| \!{\underline {\,
  1 \,}} \right. \\
\end{align}\]
From the above prime factorization, we can write
\[\Rightarrow 60=2\times 2\times 3\times 5.......\left( i \right)\]
Now, we consider the prime factorization of $72$ as shown below.
\[\begin{align}
  & 2\left| \!{\underline {\,
  72 \,}} \right. \\
 & 2\left| \!{\underline {\,
  36 \,}} \right. \\
 & 2\left| \!{\underline {\,
  18 \,}} \right. \\
 & 3\left| \!{\underline {\,
  9 \,}} \right. \\
 & 3\left| \!{\underline {\,
  3 \,}} \right. \\
 & \left| \!{\underline {\,
  1 \,}} \right. \\
\end{align}\]
From the above prime factorization we can write
$\Rightarrow 72=2\times 2\times 2\times 3\times 3.......\left( ii \right)$
From the equations (i) and (ii) we can note down the common prime factors between $60$ and $72$ as
$2$, $2$ and $3$. Now we finally multiply these to obtain the greatest common factor as
\[\begin{align}
  & \Rightarrow 2\times 2\times 3 \\
 & \Rightarrow 12 \\
\end{align}\]

Note: We must note that it is necessary to consider only the prime factors, and no composite factor for obtaining the greatest common factor. The greatest common factor is also known by the name of HCF (Highest common factor). It can also be obtained by dividing the product of the two numbers by their LCM (Least common multiple).