Prove using the notion of without loss of generality that 5x 5y is an odd integer when x and y are integers of opposite parity.

The expression is

5x + 5y

We are to prove that it is an odd integer when x and y are integers of opposite parity

First, we can assume

x = 2a (even)

y = 2b + 1 (odd)

subsituting

10 (a + b) + 5

5 [ (2 (a + b) + 1]

The term

2 (a + b) + 1 is odd and the result of an odd number multiplied by an odd number is odd