Question

Find the HCF of $12,45$ and $75$

Answer 1

Hint: In this question we have been given three numbers for which we have to find the HCF which is the highest common factor. We will solve this question by writing all the factors of the numbers and then select the factor which is the highest in all the three numbers to get the required solution.

Complete step by step answer:
The given numbers are $12,45$ and $75$.
Now to find the greatest common factor we will find the factors of the numbers.
Now the factors of $12$ using prime factorization can be found as:
$\begin{align}
  & 2\left| \!{\underline {\,
  12 \,}} \right. \\
 & 2\left| \!{\underline {\,
  6 \,}} \right. \\
 & 3\left| \!{\underline {\,
  3 \,}} \right. \\
 & \text{ }\left| \!{\underline {\,
  1 \,}} \right. \\
\end{align}$
We can see that $12$ can be written as a product of:
$12=2\times 2\times 3$
Now the factors of $45$ using prime factorization can be found as:
$\begin{align}
  & 5\left| \!{\underline {\,
  45 \,}} \right. \\
 & 3\left| \!{\underline {\,
  9 \,}} \right. \\
 & 3\left| \!{\underline {\,
  3 \,}} \right. \\
 & \text{ }\left| \!{\underline {\,
  1 \,}} \right. \\
\end{align}$
We can see that $45$ can be written as a product of:
$45=5\times 3\times 3$
Now the factors of $75$ using prime factorization can be found as:
$\begin{align}
  & 5\left| \!{\underline {\,
  75 \,}} \right. \\
 & 5\left| \!{\underline {\,
  15 \,}} \right. \\
 & 3\left| \!{\underline {\,
  3 \,}} \right. \\
 & \text{ }\left| \!{\underline {\,
  1 \,}} \right. \\
\end{align}$
We can see that $75$ can be written as a product of:
$75=5\times 5\times 3$
Now we can see that the only factor common in all the three numbers is $3$.
Since $3$is the only number that divides all the three numbers without leaving a remainder, it is the highest common factor.
Therefore, the HCF of $12,45$ and $75$ is $3$, which is the required solution.
In general form it can be written as $HCF\left( 12,45,75 \right)=3$.

Note: The greatest common factor of numbers is used for simplification purposes in an equation. It is to be remembered that for a number, $1$ and the number itself will be the factors of that number. There also exists the LCM which stands for the lowest common multiple which is the lowest multiple which two or more numbers have in common. The LCM is used to simplify fractions.