Answer to Question #92453 in Discrete Mathematics for Iula Tohi

Question

Prove the following

If a is odd and b is even, then a2 – b2 is an odd number

Answer 1

a²- b² = (a - b)(a+b)

a is odd, so a = 2k + 1, where k belongs to natural numbers

b is even, so b = 2m, where m belongs to natural numbers

So, (a - b)(a + b) = (2k + 1 - 2m)(2k + 1 + 2m) .

2k + 1 - 2m is odd, 2k+1 + 2m is odd.

Odd multiplied by odd is always equal to odd.

So, a² - b² = (a - b)(a + b) = (2k + 1 - 2m)(2k + 1 + 2m) is odd.