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Hint: First of all, try to recollect the meaning of HCF that is the highest common factor of numbers. Now prime factorize the number individually and find the factors common to all three. Now try to recollect the LCM that is the lowest common multiple of the number which is divisible by all given numbers. Take all the numbers together and prime factorize them to find the LCM of the numbers.

Here, we have to find the HCF and LCM of 6, 72, and 120 using the prime factorization method. Before proceeding with the question, let us first see what HCF and LCM are. HCF is the highest common factor between the 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. Here,

\[\begin{align}

& 4=2\times 2 \\

& 6=2\times 3 \\

& 8=2\times 2\times 2 \\

\end{align}\]

Here, HCF of 4, 6, and 8 is 2.

LCM is the lowest common multiple of the numbers. In other words, we can say that the LCM is the smallest of the least multiple common to all the numbers which is divisible by all these numbers.

For example, LCM of 12, 16 and 20 can be found as,

So, we get LCM of 12, 16, and 20 as \[2\times 2\times 2\times 2\times 3\times 5=240\]

Now, let us consider our question and find the HCF of three numbers. 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 the 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,

\[6=2\times 3\]

\[72=2\times 2\times 2\times 3\times 3\]

\[120=2\times 2\times 2\times 3\times 5\]

From the above three equations, we can see that the common factors of 6, 72, and 120 is \[2\times 3=6\]. So, we get the highest common factor or HCF of 6, 72, and 120 as 6.

Now let us find the LCM of three numbers: 6, 72, and 120. We get,

So, we get the LCM of 6, 72, and 120 as

\[2\times 2\times 2\times 3\times 3\times 5=360\]

Hence, the LCM of the three numbers is 360.

Note: In this question, students often make these mistakes while prime factorization of numbers individually or in a group, so they must note that we should always start with the smallest prime factors of the given numbers and use it until it does not become a factor of at least one number. Also, keep taking the prime factors in increasing order and don’t take them randomly to avoid making mistakes. Also, clear the confusion between HCF and LCM of two numbers, then only solve.

__Complete step-by-step solution__-Here, we have to find the HCF and LCM of 6, 72, and 120 using the prime factorization method. Before proceeding with the question, let us first see what HCF and LCM are. HCF is the highest common factor between the 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. Here,

\[\begin{align}

& 4=2\times 2 \\

& 6=2\times 3 \\

& 8=2\times 2\times 2 \\

\end{align}\]

Here, HCF of 4, 6, and 8 is 2.

LCM is the lowest common multiple of the numbers. In other words, we can say that the LCM is the smallest of the least multiple common to all the numbers which is divisible by all these numbers.

For example, LCM of 12, 16 and 20 can be found as,

So, we get LCM of 12, 16, and 20 as \[2\times 2\times 2\times 2\times 3\times 5=240\]

Now, let us consider our question and find the HCF of three numbers. 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 the 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,

\[6=2\times 3\]

\[72=2\times 2\times 2\times 3\times 3\]

\[120=2\times 2\times 2\times 3\times 5\]

From the above three equations, we can see that the common factors of 6, 72, and 120 is \[2\times 3=6\]. So, we get the highest common factor or HCF of 6, 72, and 120 as 6.

Now let us find the LCM of three numbers: 6, 72, and 120. We get,

So, we get the LCM of 6, 72, and 120 as

\[2\times 2\times 2\times 3\times 3\times 5=360\]

Hence, the LCM of the three numbers is 360.

Note: In this question, students often make these mistakes while prime factorization of numbers individually or in a group, so they must note that we should always start with the smallest prime factors of the given numbers and use it until it does not become a factor of at least one number. Also, keep taking the prime factors in increasing order and don’t take them randomly to avoid making mistakes. Also, clear the confusion between HCF and LCM of two numbers, then only solve.

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