Question

What is the greatest common factor of 54 and 36?

Answer 1

Hint: To find the greatest common factor of 54 and 36, we have to do prime factorization. Firstly, we have to list all the prime factors of 54 and 36. Then, we have to check the common factors among these (we must also consider the repeated factors also). Then, we must multiply all the common factors.

Complete step by step solution:
We have to find the greatest common factor of 54 and 36. Let us do prime factorization. Let us list all the prime factors of 54 and 36.
$\begin{align}
  & 2\left| \!{\underline {\,
  54 \,}} \right. \\
 & 3\left| \!{\underline {\,
  27 \,}} \right. \\
 & 3\left| \!{\underline {\,
  9 \,}} \right. \\
 & 3\left| \!{\underline {\,
  3 \,}} \right. \\
 & \text{ }1 \\
\end{align}$
Prime factors of 54 are $2\times 3\times 3\times 3$ .
$\begin{align}
  & 2\left| \!{\underline {\,
  36 \,}} \right. \\
 & 2\left| \!{\underline {\,
  18 \,}} \right. \\
 & 3\left| \!{\underline {\,
  9 \,}} \right. \\
 & 3\left| \!{\underline {\,
  3 \,}} \right. \\
 & \text{ }1 \\
\end{align}$
Prime factors of 36 are $2\times 2\times 3\times 3$ .
We can see that the common prime factors of 54 and 32 are
\[\begin{align}
  & 54=\fbox2\times \fbox3\times \fbox3\times 3 \\
 & 36=\fbox2\times 2\times \fbox3\times \fbox3 \\
\end{align}\]

We have to multiply all the common factors to get the HCF.
Therefore, the greatest common factors of 54 and 36 are $2\times 3\times 3=18$ .

Note: The greatest Common Factor(GCF) is also known as Highest Common Factor(HCF) or Greatest Common Divisor(GCD). We can also find the HCF using a long division method.
Firstly, we have to write the given numbers horizontally by separating it with commas. Then we have to find a prime number that exactly divides the given number. We will write this to the left. The quotient is written down.
$\begin{align}
  & 2\left| \!{\underline {\,
  54,36 \,}} \right. \\
 & 3\left| \!{\underline {\,
  27,18 \,}} \right. \\
 & 3\left| \!{\underline {\,
  9,6 \,}} \right. \\
 & \text{ }3,2 \\
\end{align}$
We will stop when we can’t divide further by the same factors.
Hence HCF of 54 and 36 are $2\times 3\times 3=18$