Question

Consider any two prime numbers and check whether the product of their HCF and LCM is equal to the product of the numbers.

Answer 1

Hint: Here we have to take any two prime numbers and we have to find both HCF i.e., highest common factor of two numbers and LCM i.e., least common multiple of the two numbers and finally we will find whether the product of the two numbers is equal to the product of HCF and LCM.

Complete step-by-step answer:
We have to consider any two prime numbers and find their HCF and LCM of the two numbers and check whether the product of their HCF and LCM is equal to the product of the numbers.
Prime numbers are the numbers that are only divided by 1 and the number itself, and prime factoring these numbers is a process of writing a number of products of its prime factors.
Let us take the two numbers as 3 and 5,
Now we find the LCM i.e., least common multiple of these two numbers, LCM of the two numbers is the least common multiple and the product of the prime factors,
Here prime numbers are 3 and 5,
So, LCM\[ = 3 \times 5 = 15\],
LCM of the two considered prime numbers is 15.
Now we have to find HCF of the numbers,
Prime numbers are the numbers that are only divided by 1 and the number itself, and prime factoring these numbers is a process of writing a number of products of its prime factors.
Again take two numbers 3 and 5,
Now we find the HCF i.e., highest common factor of these two numbers, HCF of the two numbers is the highest common factor and the product of the prime factors,
By definition, any two prime numbers will have only one common factor and that would be ‘1’, so any two different prime numbers will have the highest common factor as ‘1’, it means HCF of two prime numbers will be 1.
Here prime numbers are 3 and 5,
As factors of 3 and 5 are \[\left( {1,3} \right)\] and \[\left( {1,5} \right)\], so the only common factor to both the numbers is 1, so 1 is the HCF of the two numbers.
HCF of the two considered prime numbers is 1.
Now we have to check whether the product of their HCF and LCM is equal to the product of the numbers, i.e.,
HCF of numbers \[ \times \]LCM of Numbers = product of the numbers,
$\Rightarrow$\[1 \times 15 = 3 \times 5\],
$\Rightarrow$\[ \Rightarrow 15 = 15\],
Hence proved.

The product of their HCF and LCM is equal to the product of the numbers.

Note:
In these types of questions, students should not get confused between HCF and LCM of the numbers as LCM is a multiple of the two numbers and HCF is the factor of the numbers.