Question

Find out the HCF of the following numbers. $18,48$ .

Answer 1

Hint: To find out the HCF of the given numbers, we perform the prime factorisation of the two numbers. We find out which prime factors are common between them and how many times they are common. We see that we get one $2$ and one $3$ common. The HCF will be just the product of them.

Complete step-by-step solution:
HCF is the abbreviation or in common man’s terms short form of Highest Common Factor. By factor, we mean a number which can perfectly divide the parent number without leaving any remainder. This number will then be called the factor of the parent number. Factors are usually smaller than the parent number. There are two important facts regarding factors. The first fact is that $1$ is the factor of any number and the second fact is that a number is always a factor to itself. So, to summarize, every number has two factors at least. For two numbers, there will be one set of factors for each number. There will always be one factor common between the sets, which is $1$ and this is the lowest common factor. The largest common one will be called the HCF.
We perform prime factorisation of the two numbers. We get,
$\begin{align}
  & 2\left| \!{\underline {\,
  18 \,}} \right. \\
 & 3\left| \!{\underline {\,
  9 \,}} \right. \\
 & ~~~3 \\
\end{align}$
$\begin{align}
  & 2\left| \!{\underline {\,
  48 \,}} \right. \\
 & 2\left| \!{\underline {\,
  24 \,}} \right. \\
 & 2\left| \!{\underline {\,
  12 \,}} \right. \\
 & 2\left| \!{\underline {\,
  6 \,}} \right. \\
 & ~~~3 \\
\end{align}$
From the above prime factorisation, we get some $2's$ and $3's$ common. To be precise, we get one $2$ and one $3$ common, the product of which is $6$ .
Therefore, we can conclude that the HCF of $18,48$ is $6$ .

Note: While carrying out the prime factorisation, we must be careful to only divide by the prime numbers and not by any composite number as such. Also, before prime factorisation we should check whether out of the given numbers, one perfectly divides the other or not. If yes, then that number will itself be the HCF.