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Please prove it by full explain ation gvsnnbdb hdbdnh

Please prove it by full explain ation gvsnnbdb hdbdnh

Question Image
Grade:12th pass

1 Answers

Harsh Tiwari
54 Points
2 months ago
Given,
LCM and HCF of two numbers are the same.
Let the two rational numbers be x and y.
Given,
LCM(x,y)=HCF(x,y)
Let,
LCM(x,y)=HCF(x,y)=k, for some value k.
HCF being the highest common factor is always a factor of both the numbers.
Therefore the numbers can be written as multiples of HCF.
That is,
 x=ka , for some natural number a
 y=kb , for some natural number b
Now, since the product of two numbers is equal to the product of their LCM and HCF, we have
x×y=LCM(x,y)×HCF(x,y)
 
Substituting the values for x, y, their LCM and HCF,
ka×kb=k×k
⇒k2ab=k2
Cancelling k2 from both sides,
ab=1
⇒a=1,b=1 ( since a and b are natural numbers).
Substituting these we get x and y as
⇒x=ka=k×1=k⇒y=kb=k×1=k
⇒x=y=k
Therefore, the two numbers are the equal

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