LCM and hcf of two numbers are equal Th hen the number must be
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The LCM and HCF of two rational numbers are equal. Then the numbers are?
A) prime
B) co-prime
C) composite
D) equal
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Hint: The HCF of two numbers is always a factor of both the numbers. There is a relation between two numbers and their LCM and HCF. Using this, we can find the relation between the numbers.
Formula used: Product of two numbers is equal to the product of their Least common divisor(LCM) and Highest common factor(HCF)
x×y=LCM(x,y)×HCF(x,y)
where x and y are any two numbers.
LCM is the Least Common Multiple
HCF is the Highest Common Factor
Complete step-by-step answer:
Given data
LCM and HCF of two numbers are the same.
Let the two rational numbers be x and y.
Given that 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.
So, the correct answer is “Option D”.
Additional Information: HCF of two numbers is always less than or equal to their LCM. Also LCM is always a multiple of HCF.
Note: Two numbers are said to be co-prime when their HCF is 1. Prime numbers are those numbers with only factors 1 and the number itself. The numbers which are not prime are called composite numbers. Highest common factor (HCF) is also known as the greatest common divisor.