Question

# Find LCM and HCF of 72 and 108 by the prime factorization method.

Hint:The least number which is exactly divisible by each of the given numbers is called the least common multiple and the largest number that divides two or more numbers is the highest common factor (HCF). By using this definition, we calculate the required thing.

To find the LCM of the given numbers, we express each number as a product of prime numbers. The product of highest power of the prime numbers that appear in prime factorization of any of the numbers gives us the LCM.
To find the HCF of the given numbers, we express each number as a product of prime numbers. The highest prime factor is HCF.
Now, proceeding to our question, we are given two numbers 72 and 108.
Expanding 72 as a product of prime numbers using prime factorisation method, we get
\begin{align} & 72=2\times 2\times 2\times 3\times 3 \\ & 72={{2}^{3}}\times {{3}^{2}}\ldots (1) \\ \end{align}
Expanding 108 as a product of prime numbers using prime factorisation method, we get
\begin{align} & 108=2\times 2\times 3\times 3\times 3 \\ & 108={{2}^{2}}\times {{3}^{3}}\ldots (2) \\ \end{align}
From equation (1) and equation (2), we concluded that common factors are two 2’s and two 3’s while the extra factor is one 2 in 72 and one 3 in 108.
So, L.C.M of 72 and 108 can be expressed as:
\begin{align} & =2\times 2\times 2\times 3\times 3\times 3 \\ & =216 \\ \end{align}
H.C.F of 72 and 108 can be expressed as:
\begin{align} & =2\times 2\times 3\times 3 \\ & =36 \\ \end{align}
Thus, L.C.M and H.C.F of 72 and 108 are 216 and 36 respectively.

Note: The key step in solving this problem is the basic definition of LCM and HCF. So, by using the definition we obtained our answer. For checking the correctness of our answer, we can use a relationship: $\text{Product of two numbers = }\left( L.C.M \right)\cdot \left( H.C.F \right)$