If a, b, c and d are four positive real numbers such that sum of a, b and c is even and the  sum of b, c and d is odd, then a² - d²  is necessarily

Given:

a, b, c, and d are positive real numbers.

Sum of a, b, and c is even.

Sum of b, c, and d is odd.

We need to find the nature of a² - d².

Calculation:

1. Since the sum of a, b, and c is even, we know:

⇒ a + b + c = even ... (1)

2. Since the sum of b, c, and d is odd, we know:

⇒ b + c + d = odd ... (2)

3. Subtracting the 2nd equation from the 1st:

⇒ (b + c + d) - (a + b + c) = even - odd

⇒ a - d = odd

⇒ a and d have opposite parities (one is even, the other is odd).

In this case a + d = odd

a² - d² = (a + b)(a - b) = odd × odd = odd

∴ The correct answer is Option 3).