Question 12

The product of two consecutive integers is divisible by 2. Is this statement true or false. Give Reason?

Solution:

To determine whether the statement “The product of two consecutive integers is divisible by 2” is true or false, we can analyze the situation step by step.

1. Define Consecutive Integers:
Let the two consecutive integers be represented as $$n$$ and $$n+1$$, where $$n$$ is any integer.

2. Calculate the Product:
The product of these two consecutive integers can be expressed as:
$$P=n\times (n+1)=n^2+n$$

3. Check for Divisibility by 2:
To check if $$P$$ is divisible by 2, we need to consider the nature of $$n$$:
– If $$n$$ is even, then $$n$$ can be expressed as $$2k$$ for some integer $$k$$. Thus, the product becomes:
$$P=2k\times (2k+1)=2k(2k+1)$$
Since $$2k$$ is even, $$P$$ is divisible by 2.

– If $$n$$ is odd, then $$n$$ can be expressed as $$2k+1$$ for some integer $$k$$. Thus, the product becomes:
$$P=(2k+1)\times (2k+2)=(2k+1)\times 2(k+1)$$
Here, $$2(k+1)$$ is even, hence $$P$$ is also divisible by 2.

4. Conclusion:
In both cases (whether $$n$$ is even or odd), the product $$P$$ of two consecutive integers $$n$$ and $$n+1$$ is always divisible by 2. Therefore, the statement is true.

Final Answer:
The statement “The product of two consecutive integers is divisible by 2” is true.