Question

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

Answer 1

Let, (n – 1) and n be two consecutive positive integers
∴ Their product = n(n – 1)
= 𝑛² − 𝑛
We know that any positive integer is of the form 2q or 2q + 1, for some integer q
When n =2q, we have
𝑛² − 𝑛 = (2𝑞)² − 2𝑞
= 4𝑞² − 2𝑞
2𝑞(2𝑞 − 1)
Then 𝑛² − 𝑛 is divisible by 2.
When n = 2q + 1, we have
𝑛² − 𝑛 = (2𝑞 + 1)2 − (2𝑞 + 1)
= 4𝑞² + 4𝑞 + 1 − 2𝑞 − 1
= 4𝑞² + 2𝑞
= 2𝑞(2𝑞 + 1)
Then 𝑛² − 𝑛 is divisible by 2.
Hence the product of two consecutive positive integers is divisible by 2.