Question

Find the HCF and LCM of \[12,72\] and \[120\] by Prime Factorization method.

Answer 1

Hint: Here, we will first find the factors of all the numbers separately by the method of prime factorization and then we will find the HCF and LCM of all the given numbers by using their respective definition. Prime Factorization is a method of finding the factors of the given numbers.

Complete step-by-step answer:
We are given with the numbers 12, 72 and 120.
Now, we will find the factors of all the numbers using the method of prime factorization
First, we will find the factors of 12 using prime factorization.
 \[\begin{array}{l}3\left| \!{\underline {\,
  {12} \,}} \right. \\2\left| \!{\underline {\,
  4 \,}} \right. \\2\left| \!{\underline {\,
  2 \,}} \right. \end{array}\]
Thus the factors of 12 are \[12 = 3 \times 2 \times 2\].
Now, we will find the factors of 72 using prime factorization.
 \[\begin{array}{l}2\left| \!{\underline {\,
  {72} \,}} \right. \\2\left| \!{\underline {\,
  {36} \,}} \right. \\2\left| \!{\underline {\,
  {18} \,}} \right. \\3\left| \!{\underline {\,
  9 \,}} \right. \\3\left| \!{\underline {\,
  3 \,}} \right. \end{array}\]
Thus the factors of 72 are \[72 = 2 \times 2 \times 2 \times 3 \times 3\].
We will find the factors of 120 using prime factorization.
\[\begin{array}{l}2\left| \!{\underline {\,
  {120} \,}} \right. \\2\left| \!{\underline {\,
  {60} \,}} \right. \\2\left| \!{\underline {\,
  {30} \,}} \right. \\3\left| \!{\underline {\,
  {15} \,}} \right. \\5\left| \!{\underline {\,
  5 \,}} \right. \end{array}\]
Thus the factors of 120 are \[120 = 2 \times 2 \times 2 \times 3 \times 5\].
Thus, the factors of all the numbers are represented with the same bases as
\[\begin{array}{l}12 = {2^2} \times {3^1} \times {5^0}\\72 = {2^3} \times {3^2} \times {5^0}\\120 = {2^3} \times {3^1} \times {5^1}\end{array}\]
Now, we will find the HCF for the given numbers from the factors.
The highest common factor is a factor which is common for all the factors.
Thus, we get
HCF of \[\left( {12,72,120} \right) = {2^2} \times {3^1} \times {5^0}\]
Multiplying the terms, we get
\[ \Rightarrow \] HCF of \[\left( {12,72,120} \right) = 12\]
Now, we will find the LCM for the given numbers from the factors.
The Least Common Multiple is a multiple which is divisible by all the numbers.
Thus, we get
LCM of \[\left( {12,72,120} \right) = {2^3} \times {3^2} \times {5^1}\]
\[ \Rightarrow \] LCM of \[\left( {12,72,120} \right) = 360\]
Therefore, the HCF of (12, 72, 120) is 12 and the LCM of (12, 72, 120) is 360.

Note: The Highest Common Factor (H.C.F) of two numbers is defined as the greatest number which divides exactly both the numbers. The Least Common Multiple (L.C.M) of two numbers is defined as the smallest number which is divisible by both the numbers. HCF can be found by multiplying the factors with the least exponent common for all the factors and LCM can be found by multiplying the factors with the highest exponent from all the factors.