Question

If the sum of squares of LCM & HCF of two positive numbers is 3609 and their LCM is 57 more than their HCF, then the product of two number's is

Answer 1

Solution :-

Given that the LCM of the two numbers is 57 more than their HCF.

Let the LCM of the given two numbers be x

Then, HCF of these two numbers = x - 57

Now, according to the question.

⇒ x² + (x - 57)² = 3609

⇒ x² + x² - 114x + 3249 = 3609

⇒ 2x² - 114x - 3249 - 3609 = 0

⇒ 2x² - 114x - 360

Dividing the above by 2, we get.

⇒ x² - 57x - 180 = 0

⇒ x² - 60x - 3x - 180 = 0

⇒ x(x - 60) - 3(x - 60) = 0

⇒ (x - 60) (x - 3) = 0

⇒ x = 60 or x = 3

x = 3 is not possible because LCM of any two numbers is always greater than their HCF and HCF will always be the factor of the LCM of any given two numbers.

So, x = 60 is the correct value.

Hence, LCM of the given two numbers is 60

Then, HCF = 60 - 57

HCF = 3

Product of the two numbers = HCF*LCM

Product of the two numbers = 60*3

= 180

Product of the given two numbers is 180.

Answer.