Question

Find the HCF of $96$ and $404$ by the prime factorization method. Hence find their LCM.

Answer 1

Hint: Write both $96$ and $404$in form of the product of prime factors. Now for HCF check for the common factors present into the product of both the numbers. Evaluate the product to get HCF. For finding LCM write the product of prime factors without repeating the common ones. Now to get LCM, evaluate the product.

Complete step-by-step answer:
Here in the given problem, we are given two numbers $96$ and $404$. And we need to find the HCF of these two numbers using the prime factorization method. After this, you should find the LCM of the same numbers.
Before starting with the solution we should know about the prime factorization method. Prime factorization is a process of factoring any numbers into prime numbers. It represents a number in the form of the product of its prime factors. Prime factors are the numbers that exactly divide the given number and they are not divisible by any other number except $1$ .
Therefore for our given numbers $96$ and $404$, we have:
$ \Rightarrow 96 = 2 \times 2 \times 2 \times 2 \times 2 \times 3$ ………(i)
Similarly,
$ \Rightarrow 404 = 2 \times 2 \times 101$ ……….(ii)
Thus, the given numbers are represented using their prime factors.
Now we need to find HCF. HCF is the highest common factor or the greatest number that can divide both the numbers. This can be found by checking for the common prime factor’s product from both the numbers.
From both (i) and (ii)
$ \Rightarrow 2 \times 2$ as the common factor from both the factors of numbers
Therefore, $2 \times 2 = 4$ is the HCF of both the numbers.
Now we can find LCM for these numbers from (i) and (ii). LCM is the least or smallest common multiple of the two given numbers here. That is LCM will be a number which is exactly divisible by both these numbers and will be the smallest possible number to satisfy this.
This can be found by combining the prime factors of both numbers without any repetition.
Again from (i) and (ii), we get
$ \Rightarrow 2 \times 2 \times 2 \times 2 \times 2 \times 3 \times 101$ as the LCM of the given numbers
Therefore, $2 \times 2 \times 2 \times 2 \times 2 \times 3 \times 101 = 9696$ is the LCM of both the numbers.

Note: In questions like this understanding the terms like factor, prime number, a composite number, HCF and LCM. Prime numbers are the numbers that are not divisible by any other number except $1$ . And factors are the numbers that divide the given number and multiple is the number for which a given number is a factor. Remember that prime, LCM, HCF, composites, factors and multiples are all associated with the set of natural numbers.