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 integer, then the product is n(n – 1)

n(n – 1) = n² – n

We know that any positive integers is of the form 2q or 2q + 1 for same integer q

Case 1:

when n = 2 q

n² – n = (2q)² – 2q

= 4q² – 2q

= 2q (2q – 1)

= 2 [q(2q – 1)]

n² – n = 2 r

r = q(2q – 1)

Hence n² – n. divisible by 2 for every positive integer.

Case 2:

when n = 2q + 1

n² – n = (2q + 1)² – (2q + 1)

= (2q + 1) [2q + 1 – 1]

= 2q (2q + 1)

n² – n = 2r

r = q (2q + 1)

n² – n divisible by 2 for every positive integer.