Question

The sum of the lcm and hcf of two numbers is 1884 and their difference is 1860. if the difference between the numbers is 12, determine the sum of the numbers.

Answer 1

LCM + HCF = 1884..(1)
LCM -  HCF = 1860..(2)

Adding them,
2 LCM = 3744

LCM = 1872

HCF = 12

Let the numbers be x & y,
 
x - y =12
x= y + 12
We know,

x . y = HCF . LCM

y(y + 12) = HCF . LCM
y² + 12 y = 12 . 1872

y² + 12y - 22464 = 0

y = 144 or -156

We discard the negative answer

y =144

x = 156

x + y =300

Answer 2

Solution :-

Given :  

   L.C.M. - H.C.F. = 1860 .................(1)
   L.C.M. + H.C.F. = 1884 ................(2)

Subtracting (1) from (2),

    L.C.M. + H.C.F. = 1884 
    L.C.M. - H.C.F. = 1860
  -            +             -
_______________________
                 2H.C.F. =  24
_______________________

2H.C.F. = 24

H.C.F. = 24/2 

H.C.F. = 12

So, H.C.F. of the two numbers is 12.

Substituting this value in (1), we get.

L.C.M. - H.C.F. = 1860

L.C.M. - 12 = 1860

L.C.M. = 1860 + 12

L.C.M. = 1872

So, L.C.M of two numbers is 1872.

Let the two numbers be x and y respectively.

The difference between these two numbers is 12

⇒ x - y = 12

⇒ x = 12 + y ..................(3)

x × y = H.C.F. × L.C.M

Substituting the value of x = 12 + y in the above.

(12 + y)*(y) = 12*1872

⇒ y² + 12y = 22464

⇒ y² + 12y - 22464 = 0

⇒ y² + 156y - 144y - 22464 = 0

⇒ y(y + 156) - 144(y + 156) = 0

⇒ (y + 156) = 0 (y - 144) = 0

⇒ y = - 156  or  y = 144 

y = - 156 is a negative value and it is discarded.

Hence, y = 144

⇒ x - 144 = 12

⇒ x = 144 + 12

⇒ x = 156

Hence, two numbers are 156 and 144.

The sum of these two numbers is 156 + 144 = 300

Answer.