Question

Is it possible that HCF and LCM of two numbers be 24 and 540. Justify......
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Answer 1

Fundamental Theorem of arithmetic-

Every composite number can be expressed as a product of primes, and this factorization is unique.

There cannot be two numbers having 24 as their HCF and 540 as their LCM.

Reason :-

We know that the product of two numbers is equal to the product of their LCM and HCF.

Let the two numbers be a and b.

Then, a x b = HCF x LCM

a x b = 24 x 540

Now, for 24 to be the HCF of a and b, both a and b must be the multiple of 24.

So, let a be the smallest multiple of 24, that is 24 itself.

So, 24 x b = 24 x 540

This gives b = 540

So, a = 24 and b = 540

But, the HCF of these two numbers is 12 and not 24.

Thus, a cannot be equal to 24. Also, if we take any bigger multiple of 24, then also we will arive at a contradiction.

Thus, there cannot exist two natural numbers

having 24 as their HCF and 540 as their LCM.

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