Question

The difference of lcm and hcf of two numbers is 1572 and the sum of their lcm and hcf is 1596 if the one of the number is 144 and the other is

Answer 1

solution:-
give, one of the numbers=144
let, the other numbers=a
let, L.C.M of 144 and a be 'x'
and let the H.C.F. of144 and a be 'x'
=>x+y 1596........(¡)
and, =>x-y 1572......(¡¡)
adding the both eq.(¡) and (¡¡)
x+y =1596
x-y =1572
=2x=3168.
now, 2x =3168
=>x=3168/2=1584
x =1584.
putting the value of x=1584 in eq.(¡), we get,
x+y = 1596
=>1584+y 1596
=>y=1596-1584
y=12.
so, LCM of 144 and a =1584.
and, HCF of 144 and a =12
we know that,
LCM × HCF = product of the numbers
=>1584×12=144×a
=>144×a=1584×12
=>144a=19008
=>a=19008/144
=>a=132.
so, the other no.is 132.

Answer 2

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

One number = 144

Assumption :-

Other number be 'c'

Also,

LCM of 144 and c be 'p'

HCF of 144 and c be 'n'

p + n = 1596 ....(Equation 1)

${\boxed{\sf\:{Also\;we\;have}}}$

p - n = 1572 ...(Equation 2)  

Adding (1) and (2) we get :-

2p = 3168

${\boxed{\sf\:{p=\dfrac{3168}{2}}}}$

p = 1584

Substitute the value of p in (1),

p + n = 1596

1584 + n = 1596

n = 1596 - 1584

n = 12

LCM of 144

p = 1584

HCF is 144

And,

n = 12

Using formula we have :-

LCM × HCF = Product of the two number

1584 × 12 = 144 × c

144c = 1584 × 12

144c = 19008

${\boxed{\sf\:{c=\dfrac{19008}{144}}}}$

c = 132

Hence,

${\boxed{\sf\:{Other\;number\;=\;132}}}$