Question

Sum of 2 positive integers is 144 and their LCM is 420. Find the no.s

Answer 1

Hi there!

Let the two no.s be gx and gy,( g being their GCD)

Given, gx+gy = 144 or
g(x+y) = 144.....(1)

And, gx × gy/ g = 420 or
gxy= 420...(2)

We know, the common factor between 144 and 420 is 12. So g = 12

Putting the value in (1) and (2)

12 (x+y) = 144
=> x+y = 12...(3)

12× x × y = 420
=> xy = 35....(4)

We have, x+y = 12 and xy= 35

(x+y)^2 - 4xy = (12)^2 - 4× 35
(x-y)^2 = 144 - 140 = 4
x-y = 2....(5)

Combining equations (3) and (5)

x+y+x-y = 12 + 2
2x = 14 => x = 7

Finding y, x+ y = 12 => 7+y=12 => y =5

The numbers are - gx = 12× 7 = 84 and gy = 12× 5 = 60.

#check.

84+60=144 and LCM of 84 and 60 is 420.