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“The product of two consecutive positive integers is divisible by 2”. Is this statement true or false? Give reasons.
Given:
"Product of two consecutive positive integers is divisible by 2".
To do:
We have to find whether the given statement is true or false.
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.
Therefore, the given statement is true.Advertisements