Question

L.C.M. of the given two numbers is double the greater number. And the difference between smaller number and G.C.F. is 4. Therefore the smaller number is ............

Answer 1

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✤ Required Answer:

✒ GiveN:

• LCM is double the greater number.
• Difference between GCD and smaller number is 4.

✒ To FinD:

• Find the smaller number....?

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✤ How to Solve?

For solving this question, we need to know a simple relation between LCM, GCD and the numbers.

$\large{ \boxed{ \sf{LCM \times GCD = product \: of \: no.s}}}$

• LCM is the lowest common multiple of the numbers and GCD/HCF is the highest common factor of the two numbers.

➤ Important conditions

• $\large{ \sf{LCM \geqslant greater \: number}}$
• $\large{ \sf{GCD \leqslant smaller \: number}}$

So, By using this concepts, Let's solve the question.

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✤ Solution:

Let the numbers be a and b

• Here consider a is greater number and b is smaller number. That is a > b

We have,

• LCM = 2(Greater number)
• HCF = Smaller number - 4

That is,

➙ LCM = 2a

➙ GCD = b - 4

We know,

• LCM × HCF = product of numbers,

Then,

➙ 2a × (b - 4) = ab

➙ 2ab - 8a = ab

➙ 2ab - ab = 8a

➙ ab = 8a

➙ ab - 8a = 0

➙ a(b - 8) = 0

So,

• b = 8

☯️ Because, Greater number can't be 0, So, b = 8. Smaller number among the two 8

☀️ Hence, solved !!

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Answer 2

Answer:

$\huge{\underline{\underline{\red{\bf{Answer}}}}}$

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Given:-

• $\bold{L.C.M \ of \ two \ numbers \ is \ double \ the \ greater \ number}$
• $\bold{The \ difference \ between \ smaller \ number \ and \ G.C.F \ is \ 4}$

To find:-

• $\bold{The \ smaller \ number}$

Solution:-

$\bold{Let \ the \ smaller \ number \ be \ x \ and \ the \ greater \ number \ be \ x}$

$\bold{We \ know \ that}$

L.C.M × H.C.F=product of the numbers

Case:-1

$\bold{2y=L.C.M.}$...........(i)

Case:-2

$\bold{y \ - \ G.C.F \ = \ 4}$

$\implies$$\bold{G.C.F \ = \ y \ - \ 4}$.........(ii)

putting value in the given condition,we get

$\bold{2y \ × \ (x-4) \ = \ x \ × \ y}$

$\implies$$\bold{2xy-8y=xy}$

$\implies$$\bold{xy-8y=0}$

$\implies$$\bold{y(x-8)=0}$

$\implies$$\bold{(y-8)=0}$

$\implies$$\bold{y=8}$

•°•$\bold{Value \ of \ y \ is \ 8}$

•°•$\bold{Smaller \ number \ will \ be \ 8}$

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