Q. 24.0( 91 Votes )

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

Answer :

Let the numbers are a and a-1

Product of these number: a(a - 1) = a^{2}-a

Case 1: When a is even:

a=2p

then (2p)^{2} - 2p ⇒ 4p^{2} - 2p

2p(2p-1) ………. it is divisible by 2

Case 2: When a is odd:

a = 2p+1

then (2p+1)^{2} - (2p+1) ⇒ 4p^{2} + 4p + 1 - 2p – 1

= 4p^{2} + 2p ⇒ 2p(2p + 1) ………….. it is divisible by 2

Hence, we conclude that product of two consecutive integers is always divisible by 2

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