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

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Answer:

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

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.

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