Simple algebra: even x even = odd? Is it possible?

Question

AND If the product of two numbers is positive, then the two numbers must both be positive. Find a counterexample.

Answer 1

even x even is always even

positive product means 2 factors had same sign; -2 • -2 = +4

Answer 2

Statement: Assume x and y are even.  Then xy is even.

Proof (by contradiction).

Assume xy is odd.  It then follows that xy can be expressed in the form

xy = 2k + 1 (for some integer k).  Therefore,

xy - 1 = 2k

Since x is even and y is even, they can be expressed as

x = 2a (for some integer a)

y = 2b (for some integer b)

Substituting accordingly,

(2a)(2b) - 1 = 2k

4ab - 1 = 2k

Multiply both sides by (1/2), to get

(1/2)(4ab - 1) = k

Distribute the (1/2),

(1/2)(4ab) - (1/2) = k

2ab - (1/2) = k

Notice how 2ab is an integer, and we're subtracting (1/2), or (0.5).  This means, as a result, 2ab - (1/2) is NOT an integer.  

Therefore, k = 2ab - (1/2) is NOT an integer.

This is a contradiction (we declared k an integer from the start).

Therefore, if x and y are even, then xy must be even.

Answer 3

not possible.