Question

Find the HCF and LCM of $ 12,30{\text{ and 144}} $ by the fundamental theorem of arithmetic.
 $
  A.{\text{ HCF = 6, LCM = 720}} \\
  B.{\text{ HCF = 36, LCM = 720}} \\
  C.{\text{ HCF = 4, LCM = 720}} \\
  D.{\text{ HCF = 6, LCM = 520}} \\
  $

Answer 1

Hint: Here, first we will find prime factorization of the given numbers and then accordingly on the basis of the prime factors we will find HCF and LCM. HCF is the greatest or the largest common factor between two or more given numbers whereas the LCM is the least or the smallest number with which the given numbers are exactly divisible.

Complete step-by-step answer:
First find the prime factors of the given numbers.
Prime factorization is the process of finding which prime numbers can be multiplied together to make the original number, where prime numbers are the numbers greater than $ 1 $ and which are not the product of any two smaller natural numbers. For Example: $ 2,{\text{ 3, 5, 7,}}...... $ $ 2 $ is the prime number as it can have only $ 1 $ factor. Factors are the number $ 1 $ and the number itself.
Therefore,
 $
\Rightarrow 12 = \underline 2 \times 2 \times \underline 3 \\
\Rightarrow 30 = \underline 2 \times \underline 3 \times 5 \\
\Rightarrow 144 = \underline 2 \times 2 \times 2 \times 2 \times \underline 3 \times 3 \\
  $
The common factors in all the three given numbers are $ 2\,{\text{and 3}} $
Hence, the HCF of $ 12,30,144{\text{ is = 2}} \times {\text{3 = 6}} $

 $
\Rightarrow 12 = \underline 2 \times 2 \times \underline 3 \\
\Rightarrow 30 = \underline 2 \times \underline 3 \times 5 \\
\Rightarrow 144 = \underline 2 \times 2 \times 2 \times 2 \times \underline 3 \times 3 \\
  $
For LCM, we will write the common factors once along with the other factors.
LCM $ = 2 \times 2 \times 2 \times 2 \times 3 \times 3 \times 5 $
LCM $ = 720 $

So, the correct answer is “Option A”.

Note: Prime factorisation can also be done by the factor tree method. It is the factor-tree which shows the prime factors of the composite number in the form tree. One can get different factor trees to find the same prime factorization of the same composite number. HCF (Highest common factor is also known as the greatest common factor. LCM (Least common multiple) is also known as the least common divisor.