Question

a, b and c are even numbers and x, y and z are odd numbers. Which of the following relationships can’t be justified at any cost?
\[(a)\text{ }\dfrac{a\times b}{c}=x\times y\text{ (b) }\dfrac{a\times b}{x}=yz\text{ (c) }\dfrac{xy}{z}=ab\]
\[\begin{align}
  & (\text{A) Only B} \\
 & \text{(B) Only C} \\
 & \text{(C) All the three} \\
 & \text{(D) Only B and C} \\
\end{align}\]

Answer 1

Hint: We have to check each and every condition separately. We know that the product of two even numbers is always even. We also know that the product of two odd numbers is always odd. We also know that the product of an even number and an odd number is always even. In case a, first we will cross multiply and check whether the condition is satisfied or not. In case b and case c also we will check each and every condition.

Complete step by step answer:
In the question, it was given that a, b and c are even numbers and x, y and z are odd numbers.
Now we should check whether \[(a)\text{ }\dfrac{a\times b}{c}=x\times y\text{ }\] is justified or not.
By cross multiplication, we can write
\[(a)\text{ }a\times b\text{ }=c\times x\times y\text{ }\]
We know that the product of two even numbers is always even.
We also know that the product of two odd numbers is always odd.
We also know that the product of an even number and an odd number is always even.
So, we can say that the product of a and b is even. We can also say that the product of x and y is odd. We know that c is an even number. So, the product of x, y and c is an even number.
So, we can conclude that the product of a and b is even. We can also say that the product x, y and c is also even. So, we can say that case (a) is correct.
Now we should check whether \[\text{(b) }\dfrac{a\times b}{x}=yz\text{ }\] is justified or not.
By cross multiplication, we can write
\[(b)\text{ }a\times b\text{ }=x\times y\times z\text{ }\]
As mentioned above, the product of a and b is equal to even and the product of x and y is odd.
So, the product of x, y and z is odd.
So, we can say that case (b) is not satisfied in any condition.
Now we should check whether \[\text{(c) }\dfrac{xy}{z}=ab\] is justified or not.
By cross multiplication, we get
\[\text{(c) xy}=abz\]
We know that the product of x and y is always odd. We also know that the product of a and b is always even. So, the product of a, b and z is always even.
This statement is never correct.

So, the correct answer is “Option D”.

Note: Students may go wrong in this problem if they misconception about the product of odd numbers and even numbers. Students may have a misconception that the product of odd numbers and even numbers is an odd number. This should be avoided and concepts should be applied in a correct manner.