A positive binary number is even precisely when its last digit is 0. When is a negative 2's complement number even?

A negative 2's complement number is even precisely when its last digit is 0 as well.

Step-by-step explanation:

Proof part 1: Assume there exists a 2's complement negative even number which ends with 1. We can, therefore, express this number as

1 ... 1 = - (2's complement of 1 ... 1)

= - (0 ... 1)

≠ even

Since we know precisely that a positive binary number is not even when it ends with a 1. This is a conflict with our assumption. Our assumption is, therefore, wrong.

Proof part 2: Assume there exists a 2's complement negative odd number which ends with 0. We can, therefore, express this number as

1 ... 0 = - (2's complement of 1 ... 0)

= - (0 ... 0)

= even

Since we know precisely that a positive binary number is even when it ends with a 0. This is a conflict with our assumption. Our assumption is, therefore, wrong.