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“The product of two consecutive positive integers is divisible by 2”. Is this statement true or false? Give reasons.
"Product of two consecutive positive integers is divisible by 2".
We have to find whether the given statement is true or false.
Let the 2 consecutive numbers be, $x$ and $x\ +\ 1$.
Product $=\ x\ \times\ (x\ +\ 1)$
If $x$ is even:
Let, $x\ =\ 2k$
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:
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.
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