Question

Product of any three numbers is _______ the product of their LCM and GCD.
(A) Greater than
(B) Equal to
(C) Less than
(D) None of these

Answer 1

Hint: We solve this question by taking into consideration the concept of GCD and LCM. Then we look at some examples and find their GCD and LCM and their product. Then we compare them to find if they are greater than or equal or less than the product of the numbers. By doing so we can find the answer.

Complete step by step answer:
Let us go through the concept of LCM and GCD.
LCM of any two numbers is the least common multiple of the two numbers.
GCD or HCF of two numbers is the highest common factor of the two numbers.
We solve this question by taking an example.
Now, let us consider three numbers, say 2, 3 and 5.
Now, let us find the GCD and LCM of the numbers by factorization.
$\begin{align}
  & 2=1\times 2 \\
 & 3=1\times 3 \\
 & 5=1\times 5 \\
\end{align}$
So, we get that $GCD=1$ and $LCM=1\times 2\times 3\times 5=30$
Now, if we consider the product of given numbers we get, $2\times 3\times 5=30$.
Now, let us find the product of GCD and LCM, that is, $GCD\times LCM=1\times 30=30$.
So, we can see that the product of three numbers is equal to the product of their GCD and LCM.
Now let us consider the other three numbers 4, 6 and 5.
$\begin{align}
  & 4=1\times 2\times 2 \\
 & 6=1\times 2\times 3 \\
 & 5=1\times 5 \\
\end{align}$
So, we get that $GCD=1$ and \[LCM=1\times 2\times 2\times 3\times 5=60\].
Now, if we consider the product of given numbers we get, $4\times 6\times 5=120$.
Now, let us find the product of GCD and LCM, that is, $GCD\times LCM=1\times 60=60$.
So, we can see that the product of three numbers is greater than the product of their GCD and LCM.
We can see that the product of three numbers is not always greater than or not always equal to the product of their GCD and LCM.
So, we cannot say about them if the numbers are not given.
Hence, the answer is Option D.
Note:
The common mistake one does while solving this problem is one might consider only one type of example as we have taken first and mark the answer as Option B or consider the second example and mark the answer as Option A. So, we need to be careful while taking examples and look for all possibilities.