Question

Prove that the product of two consecutive positive integers is divisible by 2.
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Answer 1

Step-by-step explanation:

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Answer 2

$\LARGE{\textsf{\textbf{\red{Solution:-}}}}$

Let’s consider two consecutive positive integers as (n - 1) and n.

∴ Their product = (n - 1) n  = n2 – n

And then we know that any positive integer is of the form 2q or 2q + 1. (From Euclid’s division lemma for b= 2)

So, when n= 2q

We have,

⇒ n² – n = (2q)² – 2q

⇒ n² – n = 4q² -2q

⇒ n² – n = 2(2q² -q)  

Thus, n² – n is divisible by 2.

Now, when n= 2q+1

We have,

⇒ n² – n = (2q + 1)² – (2q - 1)

⇒ n² – n = (4q² + 4q + 1 – 2q+ 1)

⇒ n² – n = (4q² + 2q + 2)

⇒ n² – n = 2 (2q² + q + 1)

Thus, n² – n is divisible by 2 again.

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

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