Question

if the sum of squares of LCM and HCF of two positive number is 3609 and their LCM is 57 more than their HCF, then the product of two number is what?

Answer 1

given that the LCM of two numbers is 57 more than their HCF.
Let their LCM be x
then their HCF would be x-57
 according to the qstn,
x² +(x-57)² = 3609
x²+x²+3249-114x = 3609
2x²-114x -360 = 0
⇒ x²-57x -180 = 0
on solving we get
x= 60 or 3
3 is not possible as the LCM of any two numbers ia always greater than their HCF.
∴LCM = 60 and HCF = 60-57 = 3

LCM*HCF= product of two numbers
60*3 = 180




Hope it helps... :)

Answer 2

let LCM = l
HCF = h

l²+h²=3609

l = h+57

then
l² = h²+114h+3249

so
2h²+114h+3249 = 3609
2h²+114h-360 = 0
h²+57h-180=0
h²+60h-3h-180 = 0
(h+60)(h-3) = 0

so h = 3 as h cannot be negative

so l = 60

the no.s are 12 and 15

their product is
= 180