Question

What is the greatest common factor of 9 and 12?

Answer 1

Hint: A number can be split into other numbers called factors which when multiplied together returns the same number, when we talk about common factors, we factorize the given numbers and lookup for all the common factors. The greatest among them will be nothing but the Greatest common factor

Complete step by step solution:
We will follow certain steps to find the greatest common factor, that is to find common factors and then finding the largest among them.
Let us first factorize for 9
\[\begin{align}
  & 3\left| \!{\underline {\,
  9 \,}} \right. \\
 & 3\left| \!{\underline {\,
  3 \,}} \right. \\
 & \text{ }\left| \!{\underline {\,
  1 \,}} \right. \\
\end{align}\]
So, what we can conclude is
\[9=3\times 3\]
Similarly proceeding for 12
\[\begin{align}
  & 2\left| \!{\underline {\,
  12 \,}} \right. \\
 & 2\left| \!{\underline {\,
  6 \,}} \right. \\
 & 3\left| \!{\underline {\,
  3 \,}} \right. \\
 & \text{ }\left| \!{\underline {\,
  1 \,}} \right. \\
\end{align}\]
Simplifying it we get
\[12=2\times 2\times 3\]
Which is nothing but
\[12=3\times {{2}^{2}}\]
Note that a number can be factorized in more than one way, for example
12 can also be factorized as \[4\times 3\text{ and 6}\times 2\] but we need only prime factors to find out the greatest common factor. That is why only we used the factorization method to factorize.
Just have a look at factors of 9 and 12, there is only one factor in common in both of them which is 3 if two numbers have more than one factor in common then the largest among them is chosen which we declare to be the greatest common factor.
Since there is no other common factor, we can directly conclude that the greatest common factor of 9 and 12 is 3.

Note: While solving questions of GCF always keep in mind that GCF can never be greater than the given numbers, that is it is always less than or equal to any of two numbers. Also note that GCF can only be a prime number if you are not getting a prime number as the answer of GCF. You may have done something wrong in the solution.