Using a direct proof, how should I approach this question?

Asked Feb 27 '24 at 17:54

Active Feb 27 '24 at 17:54

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For integers $x$ and $y$, if $xy$ is even, then either $x$ or $y$ must be even.

I know how to solve this in both a contradictive and contrapositive manner but can not quite figure out how to solve it with a direct proof. An identical problem was solved via a contrapositive proof 9 years ago on this site, however even looking through that did not help me with a direct proof. I find myself having to assume that the proposition is true in order to prove it true, which I am not allowed to do.

asked Feb 27 '24 at 17:54

Arcangel0723

Use the prime factorisation of $x$ and $y$ – Vincent Batens Feb 27 '24 at 17:58
1

@VincentBatens Isn't that circular? – user469053 Feb 27 '24 at 18:03
Careful with “either $a$ or $b$,” as that often refers to the exclusive or. – Malady Feb 27 '24 at 18:21

Using a direct proof, how should I approach this question?

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