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First we need to find the prime factorization of given numbers $120$ and $144$. These both numbers are even so we can start prime factorization with prime numbers $2$.

Therefore, $120 = 2 \times 2 \times 2 \times 3 \times 5 = {2^3} \times {3^1} \times {5^1}$

Therefore, $144 = 2 \times 2 \times 2 \times 2 \times 3 \times 3 = {2^4} \times {3^2}$

Now we can see that $2$ and $3$ are common prime factors in the prime factorization of $120$ and

$144$. Also we can see the smallest powers of $2$ and $3$ are $3$ and $1$ respectively.

Therefore, HCF of $120$ and $144$ is the product of ${2^3}$ and ${3^1}$.

That is, HCF of $120$ and $144$ is ${2^3} \times {3^1} = 8 \times 3 = 24$.

Now if we observe prime factorization of both numbers then we can see that there are three prime factors $2,3$ and $5$. Also we can see the greatest powers of $2,3$ and $5$ are $4,2$ and $1$ respectively. Therefore, LCM of $120$ and $144$ is the product of \[{2^4},{3^2}\] and ${5^1}$. That is, LCM of $120$ and $144$ is \[{2^4} \times {3^2} \times {5^1} = 16 \times 9 \times 5 = 720\].

Hence, HCF and LCM of $120$ and $144$ are $24$ and $720$ respectively.