“The product of two consecutive positive integers is divisible by 2”. Is this statement true or false? Give reasons.

Given: 

"Product of two consecutive positive integers is divisible by 2".

To do: 

We have to find whether the given statement is true or false.

Solution:

Let the 2 consecutive numbers be, $x$ and $x\ +\ 1$.

Now,

Product $=\ x\ \times\ (x\ +\ 1)$

If $x$ is even:

Let, $x\ =\ 2k$

Then,

Product $=\ 2k(2k\ +\ 1)$

Product $=\ 2(2k^2\ +\ k)$

From the above equation, it is clear that the product is divisible by 2.

If  $x$  is odd:

Then,

Let,  $x\ =\ 2k\ +\ 1$

Product  $=\ (2k\ +\ 1)[(2k\ +\ 1)\ +\ 1]$

Product  $=\ (2k\ +\ 1)[2k\ +\ 2]$

Product  $=\ 2(2k^2\ +\ 3k\ +\ 1)$ 

From the above equation, it is clear that the product is divisible by 2.  

Therefore, the given statement is true.