Question

Prove that the product of two even numbers is an even number.

Answer 1

Hint: To solve this question, we need to have some knowledge on the concept of even numbers. And it is to be noted that any even number can be expressed in a general form as 2 (n), where ‘n’ is any natural number.

Complete step-by-step answer:
It is given in the question that we have to prove that the product of two even numbers is an even number. So, even numbers are those numbers which have a factor of 2 in it. Examples of even numbers are 2, 4, 6, 8, 10 …… etc. We know that 2 is the smallest even number and we can express any even number in terms of 2 (n), where ‘n’ represents any natural number. For example 12 can be expressed as 2 (6), here n = 6, similarly we can write 2 as 2 (1) here n = 1.
So, we get that the general term of an even number is 2n.
Now, let us assume two even numbers, first one as (2n) and the second one as (2m), where ‘n’ and ‘m’ represents any natural numbers.
So, the product of two even numbers, (2n) and (2m) can be written as,
$\begin{align}
  & 2n\times 2m \\
 & =\left( 4nm \right) \\
\end{align}$
We can also express (4nm) in the general form of an even number, 2 (n) as 2 (2nm), where n = 2nm.
Hence, we can say that the product of two even numbers is always an even number.

Note: The possible mistake that the students can make in this question is in the last step, where they find the product of (2n) and (2m) and get it as (4nm) and they think that (4nm) is not an even number as it is not equal to (2n) which is the general form of an even number. They get confused as there is an extra term, that is, ‘m’ in the product but this assumption is wrong because we have to express (4nm) in terms of (2n) and not equal to (2n). And we can express (4nm) in terms of (2n) by writing the value of n as 2nm.