Question

If G.C.D (a, b) = 27, L.C.M (a, b) = 729 and a > b, then a = ?
A. 729
B. 27
C. 81
D. 243

Answer 1

Hint:We will use the property of L.C.M and H.C.F or G.C.D to find the required value of a. We know that the product of L.C.M and G.C.D of any two natural numbers is equivalent to the product of the two given numbers. If we have two numbers as a and b, then, $ a\times b=G.C.D\left( a,b \right)\times L.C.M\left( a,b \right) $ .

Complete step-by-step answer:
We have been given G.C.D (a, b) = 27, L.C.M (a, b) = 729 and a > b and have been asked to find the value of a. We know that the product of L.C.M and G.C.D of any two natural numbers is equivalent to the product of the two given numbers. So, we will use this relation and find the value of b in terms of a that satisfies the given condition of a > b. So, we will check each of the options one by one.
A. 729
If we take the value of a = 729 and we apply the formula of $ a\times b=G.C.D\left( a,b \right)\times L.C.M\left( a,b \right) $ , then we will get,
  $ \begin{align}
  & 729\times b=27\times 729 \\
 & \Rightarrow b=27 \\
\end{align} $
Since a is greater than b, we can say that it satisfies the condition a > b. Hence, option A is correct.
B. 27
If we take a = 27, then applying the formula, $ a\times b=G.C.D\left( a,b \right)\times L.C.M\left( a,b \right) $ , we get,
  $ \begin{align}
  & 27\times b=27\times 729 \\
 & \Rightarrow b=729 \\
\end{align} $
We can see that the value of b is greater than a, and it does not satisfy the condition of a > b. Hence, option B is incorrect.
C. 81
If we take a = 81, and apply $ a\times b=G.C.D\left( a,b \right)\times L.C.M\left( a,b \right) $ , then we get,
  $ \begin{align}
  & 81\times b=27\times 729 \\
 & \Rightarrow b=\dfrac{27\times 729}{81} \\
 & \Rightarrow b=243 \\
\end{align} $
Now, we can again see that b is greater than a, and it does not satisfy the condition of a > b. Hence option C is also incorrect.
D. 243
If we take a = 243, then applying $ a\times b=G.C.D\left( a,b \right)\times L.C.M\left( a,b \right) $ , we get,
  $ \begin{align}
  & 243\times b=27\times 729 \\
 & \Rightarrow b=\dfrac{27\times 729}{243} \\
 & \Rightarrow b=81 \\
\end{align} $
We can see that a is greater than b and it satisfies the condition of a > b. Hence, option D is correct.
Therefore, the correct options are option A and D.

Note: The students have to remember that G.C.D is the greatest common divisor and is also known as the H.C.F or the highest common factor and L.C.M is the least common factor. The students should also remember that the product of any two natural numbers is equal to the product of their G.C.D and L.C.M. The students must try approaching the question by checking each option for the given conditions. They must not stop when they get one correct answer. They should also take note that the value provided in the options are the values of a and not b.