Foundations of Computing--- Proofing techniques HELP!?

Question

If ab+(b^2)*c is even, then b is even OR a + c is odd...

What proof and how could I use?

Answer 1

If ab+(b^2)*c is even, then b is even OR a + c is odd
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When b is even, a*b is even and also b*b*c, so the whole expression is even.

When b is odd, rewrite expression as b*(a+b*c),
hmmm.... looks like both a AND c must be odd.
Then (a+b*c) will be even and therefore the expression is even.

Answer 2

The statement is wrong, and should say that that you want a+c to be even.

The corrected statement is equivalent to:

If ab+ (b^2)*c is even AND b is odd, THEN a + c is even.

Lemma: For any non zero integer n, n*b is odd if and only if n is odd.

Proof. Left to the reader.

Lemma: For any two integers x and y, x+y is even if and only if x and y are both even or both odd.

Proof. Left to the reader.

Now, since ab + cb^2 is even, either ab and cb^2 are both even or they are both odd. But by the lemma, that means either a and c are both even or both odd. Either way, a+c is even.

Q.E.D.

The only tricky part is figuring out what facts about evenness and oddness you need to use. I went to an extreme and actually called out the two facts as "lemmas".