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