Question

Find HCF and LCM of 404 and 96 and verify that HCF $ \times $ LCM = Product of the two given numbers.

Answer 1

Hint: Write both 404 and 96 in form of the product of prime factors. Now for HCF, check for the common factors present in 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. After that multiply the HCF and LCM and compare with the product of the numbers to verify.

Complete step-by-step solution:
Here in the given problem, we are given two numbers 404 and 96. 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.
So, the prime factorization of 404 is,
$ \Rightarrow 404 = 2 \times 2 \times 101$...................….. (1)
The prime factorization of 404 is,
$ \Rightarrow 96 = 2 \times 2 \times 2 \times 2 \times 2 \times 3$...............….. (2)
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 equation (1) and (2), the common factor from both the factors of numbers are,
$ \Rightarrow 2 \times 2$
$\therefore $ HCF \[ = 4\]
Now we can find LCM for these numbers from (1) and (2). LCM is the least or smallest common multiple of the two given numbers here. That is LCM will be a number that 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.
From equation (1) and (2), the factor from both the factors of numbers without repetition is,
$ \Rightarrow 2 \times 2 \times 2 \times 2 \times 2 \times 3 \times 11$
$\therefore $ LCM \[ = 9696\]
Now multiply the HCF and LCM of the numbers,
$ \Rightarrow 4 \times 9696 = 38784$..................….. (3)
Now, find the product of the two numbers,
$ \Rightarrow 404 \times 96 = 38784$..............….. (4)
Equate equation (3) and (4),
$ \Rightarrow 38784 = 38784$
Hence, it is verified.

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.