Question

Find the HCF of 6, 72, and 120 using the prime factorization method.

Answer 1

Hint: First of all, try to recollect the HCF that is the highest common factor of numbers. Now prime factorize each number individually and find the factors common to all three.

Complete step-by-step answer:
Here, we have to find the HCF of 6, 72, and 120 using the prime factorization method. Before proceeding with the question, let us first see what HCF is. HCF is the highest common factor between numbers. In other words, we can say that HCF is the longest or the greatest factor common to any two or more given natural numbers.
For example, HCF of 4, 6, and 8 is 2. Here,
\[\begin{align}
  & 4=2\times 2 \\
 & 6=2\times 3 \\
 & 8=2\times 2\times 2 \\
\end{align}\]
Here, the highest common factor of 4, 6, and 8 is 2.

Now, let us consider the question. First of all, let us perform the prime factorization of 6, we get,

So, we get,
\[6=2\times 3....\left( i \right)\]

Now, let us perform the prime factorization of 72, we get,

So, we get,
\[72=2\times 2\times 2\times 3\times 3....\left( ii \right)\]

Now, let us perform prime factorization of 120, we get,

So, we get,
\[120=2\times 2\times 2\times 3\times 5....\left( iii \right)\]
From equation (i), (ii) and (iii), we get,
\[\begin{align}
  & 6=2\times 3 \\
 & 72=2\times 2\times 2\times 3\times 3 \\
 & 120=2\times 2\times 2\times 3\times 5 \\
\end{align}\]
We can see that the common factors of 6, 72, and 120 is \[2\times 3=6\].
So, we get the HCF of 6, 72, and 120 as 6.

Note: In this question, students often make this mistake of leaving one or another factor which is common to all three numbers. So, first of all, they should properly point out the numbers common to all the three and then only give the final answer. Also, note that during prime factorization of numbers, take each prime factor of the given number.