Question

How do you find the GCF of $30{{w}^{4}}$ , $42{{w}^{2}}$ and $12w$ ?

Answer 1

Hint: Here in this question we have been asked to find the greatest common factor of the given numbers $30{{w}^{4}}$ , $42{{w}^{2}}$ and $12w$. For answering this question, we need to find all the common factors of the given numbers and then find the greatest common factor among them.

Complete step-by-step answer:
Now considering from the question we have been asked to find the greatest common factor of the given numbers $30{{w}^{4}}$ , $42{{w}^{2}}$ and $12w$.
For answering this question, we need to find all the common factors of the given numbers and then find the greatest common factor among them.
We can say that $12w$can be given as $12\times w,3\times 4w,6\times 2w,1\times 12w,3w\times 4,6w\times 2$ . Hence the factors of $12w$ will be given as $1,2,3,4,6,12,w,2w,3w,4w,6w,12w$ .
We can say that $42{{w}^{2}}$ can be given as $\begin{align}
  & 1\times 42{{w}^{2}},w\times 42w,{{w}^{2}}\times 42,2\times 21{{w}^{2}},2w\times 21w,2{{w}^{2}}\times 21,3\times 14{{w}^{2}},3w\times 14w,3{{w}^{2}}\times 14, \\
 & 6\times 7{{w}^{2}},6w\times 7w,6{{w}^{2}}\times 7 \\
\end{align}$
Hence the factors of $42{{w}^{2}}$ will be given as $1,2,3,6,7,14,21,42,w,2w,3w,6w,7w,14w,21w,42w,{{w}^{2}},2{{w}^{2}},3{{w}^{2}},6{{w}^{2}},7{{w}^{2}},14{{w}^{2}},21{{w}^{2}},42{{w}^{2}}$.
 We can say that $30{{w}^{4}}$ can be given as $\begin{align}
  & 1\times 30{{w}^{4}},w\times 30{{w}^{3}},{{w}^{2}}\times 30{{w}^{2}},{{w}^{3}}\times 30w,{{w}^{4}}\times 30,2\times 15{{w}^{4}},2w\times 15{{w}^{3}},2{{w}^{2}}\times 15{{w}^{2}},2{{w}^{3}}\times 15w,2{{w}^{4}}\times 15, \\
 & 3\times 10{{w}^{4}},3w\times 10{{w}^{3}},3{{w}^{2}}\times 10{{w}^{2}},3{{w}^{3}}\times 10w,3{{w}^{4}}\times 10,5\times 6{{w}^{4}},5w\times 6{{w}^{3}},5{{w}^{2}}\times 6{{w}^{2}},5{{w}^{3}}\times 6w,5{{w}^{4}}\times 6, \\
 & 10\times 3{{w}^{4}},10w\times 3{{w}^{3}},10{{w}^{2}}\times 3{{w}^{2}},10{{w}^{3}}\times 3w,10{{w}^{4}}\times 3 \\
\end{align}$Hence the factors of $30{{w}^{4}}$ will be given as $\begin{align}
  & 1,2,3,5,6,10,15,30,w,2w,3w,5w,6w,10w,15w,30w,{{w}^{2}},2{{w}^{2}},3{{w}^{2}},5{{w}^{2}},6{{w}^{2}},10{{w}^{2}},15{{w}^{2}},30{{w}^{2}}, \\
 & {{w}^{3}},2{{w}^{3}},3{{w}^{3}},5{{w}^{3}},6{{w}^{3}},10{{w}^{3}},15{{w}^{3}},30{{w}^{3}},{{w}^{4}},2{{w}^{4}},3{{w}^{4}},5{{w}^{4}},6{{w}^{4}},10{{w}^{4}},15{{w}^{4}},30{{w}^{4}} \\
\end{align}$.
Now we can say that the common factors are $1,2,3,6,w,2w,3w,6w$ .
Therefore we can conclude that the greatest common factor of the given numbers $30{{w}^{4}}$ , $42{{w}^{2}}$ and $12w$is given as $6w$.

Note: While answering questions of this type we should be sure with the calculations that we are going to perform and the concepts that we are going to apply in between the steps for simplifying the given algebraic expression. This is a very simple and easy question and can be answered accurately in a short span of time. Very few mistakes are possible in questions of this type. Alternatively we can answer this question using prime factorization method as follows
$\begin{align}
  & w\left| \!{\underline {\,
  \begin{matrix}
   12w, & 42{{w}^{2}}, & 30{{w}^{4}} \\
\end{matrix} \,}} \right. \\
 & 6\left| \!{\underline {\,
  \begin{matrix}
   12, & 42w, & 30{{w}^{3}} \\
\end{matrix} \,}} \right. \\
 & \left| \!{\underline {\,
  \begin{matrix}
   2, & 7w, & 5{{w}^{3}} \\
\end{matrix} \,}} \right. \\
\end{align}$
Hence we can say that the greatest common factor will be given as $6w$ .