Question

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

Answer 1

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.