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Prove that the product of two consecutive positive integers is divisible by 2.
Given: Statement "Product of two consecutive positive integers is divisible by 2".
To prove: Here we have to prove the given statement.
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.
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