Proof that the product of two consecutive positive integers is divisible by 2
1. Consecutive integers:
When we have two consecutive positive integers, they can be represented as n and n+1, where n is any positive integer.
2. Product of two consecutive integers:
The product of these two consecutive integers is n*(n+1).
3. Even and odd integers:
One of the consecutive integers is always even and the other is always odd. This is because an even number plus or minus 1 results in an odd number.
4. Divisibility by 2:
Since one of the consecutive integers is always even, it can be expressed as 2k, where k is an integer. Therefore, the product n*(n+1) can be rewritten as 2k*(2k+1).
5. Divisibility by 2:
When we expand 2k*(2k+1), we get 4k^2 + 2k. This expression clearly contains a factor of 2, making it divisible by 2.
6. Conclusion:
Therefore, the product of two consecutive positive integers (n*(n+1)) is always divisible by 2 because one of the integers is even, leading to a factor of 2 in the product.
By following the logic above, we have proven that the product of two consecutive positive integers is always divisible by 2.