Question

Find the HCF of $135$ and $75$ by the prime factorization, hence find the LCM and HCF of $135$, $75$ and $20$.

Answer 1

Hint: To solve the given question, we will first find out what LCM and HCF of numbers are. Then, we will find the HCF of $135$ and $75$ by the prime factorization method. In this method, we will write both the numbers as the product of prime numbers. Then, we will select the numbers which are common to both the numbers and multiply them to get the HCF. The LCM of these two will be obtained by dividing the product of numbers by HCF. Then, to find the HCF of $135$, $75$ and $20$, we will calculate the HCF of $135$, $75$ and $20$. Similarly, we will find the LCM.

Complete step-by-step answer:
To start with, we will first find out what LCM and HCF are. HCF of two or more numbers is the greatest positive integer that divides each of the integers. LCM of two or more numbers is the smallest positive integer that is divisible by each of the numbers.
Now, we will find the HCF of $135$ and $75$ by the prime factorization method. In this method, we write $135$ and $75$ as the product of prime numbers. Thus, we have,
\[135=3\times 3\times 3\times 5\]
\[75=3\times 5\times 5\]
Now $3$ and $5$ are common to both, so HCF will be the product of $3$ and $5$. Thus, we have,
\[\text{HCF of 135 and 75}=3\times 5\]
\[\Rightarrow \text{HCF of 135 and 75}=15\]
Now, we will make use of the product of HCF and LCM of two numbers equal to the product of the numbers. With the help of this, we will find the LCM of $135$ and $75$ . Thus, we have,
\[\left( HCF \right)\times \left( LCM \right)=\text{Product of numbers}\]
\[15\times \left( LCM \right)=135\times 75\]
\[\Rightarrow LCM=\dfrac{135\times 75}{15}\]
\[\Rightarrow LCM=675\]
Thus, the $LCM$of $135$ and $75$ is $675$ .
Now, we have to calculate the HCF of $135$, $75$ and $20$. Now, we will make use of the fact that HCF of $135$, $75$ and $20$will be equal to the HCF of HCF $\left( 135,75 \right)$ and $20$.
Now, we will have to find the HCF of $15$ and $20$ by the prime factorization method.
Therefore, we can say that,
\[15=3\times 5\]
\[20=2\times 2\times 5\]
Thus, the HCF of $15$ and $20$ will be $5$ as only $5$ is common. Thus, we have,
\[HCF\left( 15,20 \right)=5\]
\[HCF\left( 135,75,20 \right)=5\]
Similarly, the LCM of $135$, $75$ and $20$will be equal to the LCM of LCM $\left( 135,75 \right)$ and $20$. Thus, we have to calculate the LCM of $675$ and $20$ :
Now, we have the prime factors as follows:
\[675=3\times 3\times 3\times 5\times 5\]
\[20=2\times 2\times 5\]
The HCF of $675$ and $20$ will be $5$ as only $5$ is common. Now, we know that,
\[\left[ LCM\left( 675,20 \right) \right]\times \left[ HCF\left( 675,20 \right) \right]=675\times 20\]
\[\Rightarrow \left[ LCM\left( 675,20 \right) \right]\times 5=675\times 20\]
\[\Rightarrow LCM\left( 675,20 \right)=\dfrac{675\times 20}{5}\]
\[\Rightarrow LCM\left( 675,20 \right)=2700\]
\[\Rightarrow LCM\left( 135,75,20 \right)=2700\]
Therefore, the LCM is $2700$

Note: Students can make the mistakes while taking out the prime factors as the numbers given are a bit larger so the number of factors will be more so students may get confused. The dimension of using LCM of two numbers starts with basic math operations such as addition and subtraction on fractional numbers. In math problems where we pair two objects against each other, the LCM value is useful in optimizing the quantities of the given objects.