Step by step
The question
Question: prove that if n³ is even, then n 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.
Turned around, it says: if n is odd, then n³ is odd.
Starting from n is much easier: we know how to write it with letters.
Careful!
That swaps the two parts.
It is the converse, a different statement.
Turned around means: swap the parts and put not on both.
2 Write it with letters
n is odd, so n = 2a + 1, where a is a whole number.
That is what odd means.
3 Put them in
Now put them in: n³ becomes (2a + 1)³.
4 Work it out
(2a + 1)³ means the bracket times itself.
Every part times every part: 8a³ + 4a² + 4a² + 2a + 4a² + 2a + 2a + 1.
Like terms together: 8a³ + 12a² + 6a + 1.
5 Show the form
We take out 2 from all but the 1: 2(4a³ + 6a² + 3a) + 1.
4a³ + 6a² + 3a is a whole number, because whole numbers added and multiplied stay whole.
2 times a whole number, plus 1: that is the form of an odd number.
So n³ is odd.
The turned statement is true, so the first one is true too: if n³ is even, then n is even.
The square at the end means: the proof is done.
6 Check
Try n = 2.
Then n³ is 8: even.
And 2 is even, as the statement says.
Right!
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.