Question

The LCM and HCF of two rational numbers are equal. Then the numbers are?
A) prime
B) co-prime
C) composite
D) equal

Answer 1

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)
$$x\times y=LCM(x,y)\times HCF(x,y)$$
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
$$x\times y=LCM(x,y)\times HCF(x,y)$$
Substituting the values for $x$, $y$, their LCM and HCF,
$ka\times kb=k\times k$
$\Rightarrow k_{}^{2}ab=k_{}^2$
Cancelling $k_{}^2$ from both sides,
$ab=1$
$\Rightarrow a=1,b=1$ ( since $a$ and $b$ are natural numbers).
Substituting these we get $x$ and $y$ as
$$\begin{gathered} \Rightarrow x=ka=k\times 1=k \\ \Rightarrow y=kb=k\times 1=k \end{gathered}$$
$\Rightarrow x=y=k$
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.