Proof that if 32 -|- ((a^2+3)(a^2+7)) if a is even

[mhz]

Homework Statement

Proof that if 32 -|- ((a^2+3)(a^2+7)) if a is even. (note: -|- means NOT divides)

The question asks for a proof of this statement.

Homework Equations

If a | b, then there exists an integer k such that b = ka.

The Attempt at a Solution

If the contrapositive of the statement is true, then so is the statement. So we will prove that if a is odd, then 32 | ((a^2+3)(a^2+7)).Since a is odd, (a^2+3)(a^2+7) is even, 2 | ((a^2+3)(a^2+7)). This must mean that there exists an integer k such that ((a^2+3)(a^2+7)) = 2k. ...

 

[dextercioby]

32 can't divide the product of 2 odd numbers, right ?

 

[mhz]

Right, but I'm trying to prove that in a sort of fashion that I've already shown (long-winded and specifically showing each step, trying not to assume much).

What you're arguing is something like:

Suppose a is even. Then a^2 is even, so both a^2+3 and a^2+7 are odd. Since 32 is even, 32 \nmid ((a^2+3)(a^2+7)).

However that logic is flawed. This is saying that since the converse of the statement is true, so too must the statement be true. (but that is wrong)

 

[dextercioby]

You are in error. The proof you posted which you read from my statement is perfecty ok. There's no converse anywhere. The converse would have been something like:

Prove that if 32 doesn't divide (a^2 +3) (a^2 +7) then a is even.

As you can see the converse in not involved in the proof you posted.

 

[mhz]

Prove that 32 divides (a^2 +3) (a^2 +7) if a is odd.

That is the contrapositive, not the converse.

Converse of "if A then B" is "if B then A", the contrapositive is "if not B then not A".

Since A in this case is "if 32 doesn't divide (a^2 +3) (a^2 +7)" and B is "a is even". The converse would be "if a is even then 32 doesn't divide (a^2 +3) (a^2 +7)" and the proof I posted in my second post proves that. However, my point is that the proof of the converse does not always prove the statement.

What I'm trying to do is one of the following:
a) Prove the statement: "if 32 doesn't divide (a^2 +3) (a^2 +7) then a is even"
b) Prove the contrapositive: "if a is odd, 32 divides (a^2 +3) (a^2 +7)"

A little while after my last post I came up with this, and I think this proof is ok:

If the contrapositive of the statement is true, then so is the statement. So we will prove that if a is odd, then 32 | ((a^2+3)(a^2+7)).

Since a is odd, a^2 must be odd so both a^2 +3 and a^2 +7 must be even, but also since a^2 ≥ 1, ((a^2+3)(a^2+7)) ≥ 4*8 ≥ 32.
Therefore 32 | ((a^2 +3)(a^2 + 7)).

 

[jbunniii]

Proof that if 32 -|- ((a^2+3)(a^2+7)) if a is even. (note: -|- means NOT divides)

I think the confusion is stemming from the fact that you have two "ifs" in your problem statement. One of them should not be there, and the meaning will be different depending on which one you keep.