Prove if xy is even, then x is even or y is even Step by Step

Step by step

The question

Question: prove that if xy is even, then x is even or y is even.

1 Turn it around

An if-then statement means the same as its contrapositive: if not Q, then not P.

We saw it with truth tables.

if P, then Q ≡ if not Q, then not P

Turned around, it says: if x and y are both odd, then xy is odd.

Starting from x and y is much easier: we know how to write them with letters.

turned around: if x and y are both odd, then xy is odd

Careful!

That swaps the two parts.

It is the converse, a different statement.

Turned around means: swap the parts and put not on both.

Careful: not if x is even or y is even, then xy is even ✗

2 Write it with letters

x is odd, so x = 2a + 1, where a is a whole number.

x = 2a + 1

That is what odd means.

odd = 2 × a whole number + 1

y is odd, so y = 2b + 1, where b is a whole number.

A new letter, b: y can be a different number from x.

y = 2b + 1

3 Put them in

Now put them in: xy becomes (2a + 1)(2b + 1).

xy = (2a + 1)(2b + 1)

4 Work it out

Every part times every part: 4ab + 2a + 2b + 1.

= 4ab + 2a + 2b + 1

5 Show the form

We take out 2 from all but the 1: 2(2ab + a + b) + 1.

= 2(2ab + a + b) + 1

2ab + a + b is a whole number, because whole numbers added and multiplied stay whole.

2ab + a + b is a whole number

2 times a whole number, plus 1: that is the form of an odd number.

So xy is odd.

The turned statement is true, so the first one is true too: if xy is even, then x is even or y is even.

The square at the end means: the proof is done.

So: if xy is even, then x is even or y is even. ∎

6 Check

Try x = 2 and y = 5: it gives 10.

x = 2, y = 5: xy = 10, even

10 is even, and 2 is even, as the statement says.

Right!

and 2 is even

Let's look back.

First: turn it around.

Then: write it with letters.

Then: put them in.

Then: work it out.

Then: show the form.

Last: check.

Answer