Prove That the Product of Two Consecutive Positive Integers Is Divisible by 2

Video Explanation

Question

Prove that the product of two consecutive positive integers is divisible by 2.

Solution

Step 1: Let the Two Consecutive Positive Integers Be

Let the first positive integer be \(n\), where \(n\) is a positive integer. Then the next consecutive integer is \(n+1\).

So the two consecutive positive integers are:

\[ n \quad \text{and} \quad n+1 \]

Step 2: Write the Product

The product of these two integers is:

\[ n(n+1) \]

Step 3: Show That the Product Is Divisible by 2

We know that among any two consecutive integers, one must be even. That means either:

• \(n\) is even, or
• \(n+1\) is even.

An even number is always divisible by 2. Since one of the factors of the product \(n(n+1)\) is even, the entire product is divisible by 2.

Therefore:

\[ n(n+1) \quad \text{is divisible by } 2. \]

Conclusion

Hence, the product of two consecutive positive integers is divisible by 2.

\[ \therefore \quad \text{Proved.} \]