Question

Find the HCF of 12, 18 and 24.

Answer 1

Hint: We will write the prime factorization of each number. Then we will make a list of factors that are common to all three numbers. The highest common factor is the product of all the common factors of the given numbers. We will multiply the common factors that we listed together to obtain the highest common factor of the given numbers.

Complete step by step answer:
The given numbers are 12, 18 and 24. Let us look at the prime factorization of 12. We can factorize 12 in its prime factors as follows,
 $ \begin{align}
  & 2\left| \!{\underline {\,
  12 \,}} \right. \\
 & 2\left| \!{\underline {\,
  6 \,}} \right. \\
 & 3\left| \!{\underline {\,
  3 \,}} \right. \\
 & \left| \!{\underline {\,
  1 \,}} \right. \\
\end{align} $
Next, we will see the prime factorization of the number 18. The number 18 can be factorized into its prime components in the following manner,
 $ \begin{align}
  & 2\left| \!{\underline {\,
  18 \,}} \right. \\
 & 3\left| \!{\underline {\,
  9 \,}} \right. \\
 & 3\left| \!{\underline {\,
  3 \,}} \right. \\
 & \left| \!{\underline {\,
  1 \,}} \right. \\
\end{align} $
The next number is 24. The prime factorization of 24 is as follows,
 $ \begin{align}
  & 2\left| \!{\underline {\,
  24 \,}} \right. \\
 & 2\left| \!{\underline {\,
  12 \,}} \right. \\
 & 2\left| \!{\underline {\,
  6 \,}} \right. \\
 & 3\left| \!{\underline {\,
  3 \,}} \right. \\
 & \left| \!{\underline {\,
  1 \,}} \right. \\
\end{align} $
Now, we will make a list of the prime factors that are common in the factorizations of all three numbers. So, the common factors from the above three factorizations are 2 and 3. We know that the highest common factor of the given numbers is the product of all the common factors of the given numbers. So, the highest common factor of 12, 18, and 24 is $ 2\times 3=6 $ .

Note:
It is beneficial to write the factorizations of numbers into their prime factors explicitly. This will enable us to avoid making any minor mistakes such as repetitions or omissions of factors. We should be aware of the difference between the highest common factor and the least common multiple. The least common multiple is the number that can be divided by all the given numbers and the highest common factor is the highest number that divides all the given numbers.