Question

Find HCF and LCM of 120 and 144 by using Fundamental theorem of Arithmetic.

Answer 1

Hint: Fundamental Theorem of Arithmetic states that every composite number can be expressed as a product of prime factors.
Prime numbers are numbers that are divisible only by 1 and itself. First, express both the numbers as a product of its prime factors.
Then, to find LCM, multiply the common factor with the highest power and all the remaining factors. To find HCF, take the common factors with the lowest power and multiply them.

Complete step by step answer:
 First, express both the numbers as a product of prime factors.
120 can be expressed as:
\[\begin{align}
  & 120=2\times 2\times 2\times 3\times 5 \\
 & 120={{2}^{3}}\times 3\times 5 \\
\end{align}\]
144 can be expressed as:
\[\begin{align}
  & 144=2\times 2\times 2\times 2\times 3\times 3 \\
 & 144={{2}^{4}}\times {{3}^{2}} \\
\end{align}\]
The factors with highest power among the common factors are \[{{2}^{4}}\] and \[{{3}^{2}}\]. Multiply both with the remaining factor, 5 to find LCM. Therefore,
\[\text{LCM}={{2}^{4}}\times {{3}^{2}}\times 5=720\]
The common factors with the lowest power are \[{{2}^{3}}\] and \[{{3}^{1}}\]. Multiply both to find HCF. Therefore,
\[\text{HCF}={{2}^{3}}\times 3=24\]

Hence, LCM of 120 and 144 is 720 and HCF is 24.

Note: We should not get confused in calculating HCF and LCM. An easy way to remember which factors to choose is that since HCF is the ‘highest’ common factor, factors with ‘lowest’ power are to be selected and since LCM is ‘lowest’ common multiple, factors with ‘highest’ power are to be selected along with the remaining factors.
If there are no common factors between two numbers, the H.C.F will be 1 and LCM will be the product of both numbers. Another way of finding HCF and LCM is through a division method.