Question

What is the greatest common factor for 12 and 15?

Answer 1

Hint: To obtain the greatest common factor of two numbers we have to split each number in there factors which when multiplied again give the same number. We factorize both numbers and look for common factors in between them and the product of all common factors in the given number is known as the greatest common factor.

Complete step by step solution:
To find the greatest common factor of 12 and 15 we will firstly find the factor of both by using factorization method.
For finding the factor of 12 we will use factorization method as below:
$\begin{align}
  & 2\left| \!{\underline {\,
  12 \,}} \right. \\
 & 2\left| \!{\underline {\,
  6 \,}} \right. \\
 & 3\left| \!{\underline {\,
  3 \,}} \right. \\
 & \left| \!{\underline {\,
  1 \,}} \right. \\
\end{align}$
So we get,
$12=2\times 2\times 3$
We can rewrite it as,
$12={{2}^{2}}\times 3$………$\left( 1 \right)$
In Similar way we will find the factor of 15 by using factorization method as below:
$\begin{align}
  & 3\left| \!{\underline {\,
  15 \,}} \right. \\
 & 5\left| \!{\underline {\,
  5 \,}} \right. \\
 & \left| \!{\underline {\,
  1 \,}} \right. \\
\end{align}$
So we get,
$15=3\times 5$………$\left( 2 \right)$
Now, if we compare equation (1) and (2) we get that there is only one factor common in both which is 3.

Hence the greatest common factor of 12 and 15 is 3.

Note: The Factorization method is used when the greatest common factor is to be calculated for given numbers because GCF is always a prime number and only this method gives the prime factors of a number. If we are not getting a prime number as our GCF that means we have made some mistake in the calculation part. One thing that should be kept in mind while calculating the greatest common factor is that it can never be bigger than the given numbers.