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

Question

Philoid · Accepted Answer

Let n and n-1 be the 2 positive integers.

Product = n × (n-1)

= n² – n

Case I (when n is even):

Let n = 2q

n² – n = (2q) ² - 2q

= 4q² - 2q

= 2q × (2q-1)

Hence the product n² – n is divisible by 2

Case II (when n is odd):

Let n be 2q + 1

n² – n = (2q + 1)²- (2q + 1)

= 4q² + 4q + 1 - 2q – 1

= 4q² + 2q

= 2q × (2q + 1)

Hence the product n² – n is divisible by 2

Hence Proved