Question

Find the HCF and LCM of $18$ and $45$ by prime factorisation method.

Answer 1

Hint:
We will factorise $18$ and $45$ individually and then we will calculate the HCF and LCM of $18$ and $45$. For HCF, we will multiply the common factors of $18$ and $45$ . For LCM, we will multiply the factors of $18$ and $45$ individually (without repeating a factor if common in both $18$ and $45$ for e.g., if $18$ = $a \times b \times b$ and $45$ = $b \times b \times c$, then the LCM of $18$ and $45$ will be $a \times b \times b \times c$).

Complete step by step solution:
We are given two numbers $18$ and $45$. We are required to calculate their HCF and LCM by prime factorisation method.
Definition: In mathematics, the prime factorisation method is the way of expressing a certain number in terms of the product of its prime factors. Or, we can say that it is a method of breaking down the given integer into its prime factors.
The prime factors of $18$ are: $18 = 2 \times 3 \times 3$
The prime factors of $45$ are: $45 = 3 \times 3 \times 5$
For determining the HCF of $18$ and$45$, we will multiply the common prime factors. Here, $3 \times 3$ is common in the prime factorization of both $18$ and$45$. So, we can say that the Highest Common Factor (HCF) of $18$ and $45$ can be obtained as:
$ \Rightarrow {\text{HCF}}\left( {18,45} \right) = 3 \times 3 = 9$
Therefore, the HCF of $18$ and$45$ is $9$.
To obtain the LCM of $18$ and$45$, we will multiply all the factors of $18$ and$45$(without repeating the common factors in individual prime factorization). Here, $3 \times 3$ is the common factor in both $18$ and $45$hence, we will multiply it only once . So, the Least Common Multiple (LCM) of $18$ and $45$ can be given by:
$ \Rightarrow {\text{LCM}}\left( {18,45} \right) = 2 \times 3 \times 3 \times 5 = 90$

Therefore, the HCF of $18$ and $45$ is $90$.

Note:
In this question, you may get confused in the determination of LCM since you have to avoid multiplying $3 \times 3$ twice as it is common in the factorization of both $18$ and$45$. HCF is defined as the greatest number which divides each of the two numbers (or more) and that’s why we have multiplied the common factors while LCM is defined as the smallest number which is divisible by both numbers (or more) whose LCM is required to find.