Question

The product of the LCM and
HCF of two numbers is 24. -
The differnce of the numbers
is two. find the two numbers​

Answer 1

AnsweR :

$\bf{\Large{\underline{\sf{Given\::}}}}$

The product of the L.C.M. and H.C.F. of two numbers is 24. The difference of the number is 2.

$\bf{\Large{\underline{\sf{To\:find\::}}}}$

The two number.

$\bf{\Large{\underline{\rm{\red{Explanation\::}}}}}$

Assume the two number be R and M.

A/q

$\leadsto\sf{R-M=2}\\\\\leadsto\sf{\purple{R=2+M....................(1)}}$

&

$\implies\sf{L.C.M.*H.C.F.=24}$

Or

$\implies\sf{RM\:=\:24}\\\\\\\implies\sf{(2+M)(M)=24\:\:\:\:\:\:\:\:\:\:\:\bigg[R\:=\:2+M\bigg]}\\\\\\\implies\sf{2M+M^{2} =24}\\\\\\\implies\sf{M^{2} +2M-24=0}\\\\\\\implies\sf{M^{2} +6M-4M-24=0}\\\\\\\implies\sf{M(M+6)-4(M+6)=0}\\\\\\\implies\sf{(M+6)(M-4)=0}\\\\\\\implies\sf{M+6=0\:\:\:\:\:Or\:\:\:\:\:M-4=0}\\\\\\\implies\sf{\red{M\:=\:-6\:\:\:\:\:Or\:\:\:\:\:\:M\:=\:4}}$

∴We know that negative value is not acceptable.

Hence,

M = 4

____________________________

$\longmapsto\sf{R\:=\:2+4}\\\\\\\longmapsto\sf{\red{R\:=\:6}}$

$\bigstar$ The two number is 6 and 4.

Answer 2

Answer;

$\sf{the \: product \: of \: lcm \: and \: hcf \: of \: two}$

$\sf{number \: is \: alwayz \: equal \: to \: the \: product \: of \: the \: number }$

$\rm{let \: number \: be \: x}$

$\rm{second \: number \:be \: x + 2}$

$\sf{ {x}^{2} (x + 2) = 24}$

$\sf{i.e \: (x + 6)(x - 4)} = 0$

$\rm{either}$

$\sf{(x + 6 = 0) \: (x - 6 = 0)}$

___________________________

$\sf{ or \: ( x - 4 = 0) \: ( x + 4 = 0)}$

$\sf{ \boxed{ \red{ \sf{numbers \: are \: positive \: }}}}$

$\rm{then \: second \: number}$

$\sf{ = x + 2 = 4 + 2 = 6}$

$\sf{4 \: and \: 4 + 2 = 6}$

so number are 4 and 6

$\rm{ \boxed{ \red{ \rm{4 \: and \: 6}}}}$