Question

What is the GCF of 16, 24 and 32?

Answer 1

Hint: To obtain the greatest common factor of the given three numbers we will use factoring method. Firstly we will find the prime factor of all the three numbers then we will simplify their factors by adding the power of the same term. Then we will look for common factors in all of them and multiply the common factor to get the desired answer.

Complete step-by-step answer:
The three numbers given are as follows:
$16,24,32$
Now we will find prime factor of each number separately as:
For number 16 we get,
$\begin{align}
  & 2\left| \!{\underline {\,
  16 \,}} \right. \\
 & 2\left| \!{\underline {\,
  8 \,}} \right. \\
 & 2\left| \!{\underline {\,
  4 \,}} \right. \\
 & 2\left| \!{\underline {\,
  2 \,}} \right. \\
 & 1 \\
\end{align}$
So the prime factors of 16 are:
$36=2\times 2\times 2\times 2$
On simplifying more we can write the above equation as:
$36={{2}^{4}}$…..$\left( 1 \right)$
Next, for number 24 we will get,
$\begin{align}
  & 2\left| \!{\underline {\,
  24 \,}} \right. \\
 & 2\left| \!{\underline {\,
  12 \,}} \right. \\
 & 2\left| \!{\underline {\,
  6 \,}} \right. \\
 & 3\left| \!{\underline {\,
  3 \,}} \right. \\
 & 1 \\
\end{align}$
So the prime factors of 24 are:
$24=2\times 2\times 2\times 3$
On simplifying more we can write above equation as:
$24={{2}^{3}}\times 3$…..$\left( 2 \right)$
For number 32 we get,
$\begin{align}
  & 2\left| \!{\underline {\,
  32 \,}} \right. \\
 & 2\left| \!{\underline {\,
  16 \,}} \right. \\
 & 2\left| \!{\underline {\,
  8 \,}} \right. \\
 & 2\left| \!{\underline {\,
  4 \,}} \right. \\
 & 2\left| \!{\underline {\,
  2 \,}} \right. \\
 & 1 \\
\end{align}$
So the prime factors of 36 are:
$36=2\times 2\times 2\times 2\times 2$
On simplifying more we can write the above equation as:
$36={{2}^{5}}$…..$\left( 3 \right)$
On comparing equation (1) (2) and (3) we get common factors as:
Common factor $={{2}^{3}}$
Therefore,
Greatest common factor $=8$
Hence GCF of 16, 24 and 32 is 8.

Note: GCF also known as greatest common factor is the largest factor of the given numbers which means it is the largest numbers that divide all the given numbers completely. Another method to find the GCF is by finding all the factor of the given number and taking out the common factor among them and the highest factor from the common factor will be our GCF.