Solution to Prove: If ab is even then a or b is even. - Sikademy
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Archangel Macsika

Prove: If ab is even then a or b is even.

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As i understand, it is implied that numbers a and b must be integer(otherwise, the statement is false). So, this numbers can be even or odd. There are 4 possibilities:

1).a, b are both even

2).a, b are both odd

3).a is even, b is odd

4).a is odd, b is even

Firstly we will examine (2) condition: a and b are both odd means a=2n-1b=2k-1 where n and k both \in Z(+)

Then ab=(2n-1)(2k-1)=4nk - 2(n+k) +1

4nk and 2(n+k) are both even, then their sum is even, then their sum plus 1 is odd. ab is odd.

We considered (2) situation and came to conclusion that: (2)\to (ab is odd). That means: \lnot (ab is odd) \to \lnot (2)

\lnot (ab is odd) means (ab is even)

(ab is even) \to \lnot (2).

So, if ab is even than at least one of the two numbers is even

Statement has been proven

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