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

$\textbf{\underline{\underline{According\:to\:the\:Question}}}$

LCM of the two numbers

= 57

It is more than their HCF

Assume :-

LCM of the given two numbers be p

HCF of two numbers

= p - 57

$\textbf{\underline{Situation}}$

p² + (p - 57)² = 3609

p² + p² - 114p + 3249 = 3609

2p² - 114p - 3249 - 3609 = 0

2p² - 114p - 360 .... (1)

${\boxed{\sf\:{Dividing\;(1)\;by\;2}}}$

p² - 57p - 180 = 0

p² - 60p - 3p - 180 = 0

p(p - 60) - 3(p - 60) = 0

(p - 60) (p - 3) = 0

p = 60

p = 3

p = 3 (It is not possible)

Hence,

p = 60

LCM of the given two numbers = 60

Therefore,

HCF = 60 - 57

HCF = 3

$\textbf{\underline{Using\;Formula\;we\;have}}$

HCF × LCM = Product of the two numbers

60 × 3 = Product of the two numbers

180 = Product of the two numbers

$\textbf{\underline{We\;Get}}$

= 180