Step by step
The question
Question: prove that the product of two odd numbers is odd.
1 Write it with letters
x is odd, so x = 2a + 1, where a is a whole number.
That is what odd means.
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.
Careful!
The same letter would make x and y the same number.
Two numbers need two letters.
2 Put them in
Now put them in: xy becomes (2a + 1)(2b + 1).
3 Work it out
Every part times every part: 4ab + 2a + 2b + 1.
4 Show the form
We take out 2 from all but the 1: 2(2ab + a + b) + 1.
2ab + a + b 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 xy is odd.
The square at the end means: the proof is done.
5 Check
An example: x = 3 and y = 5.
It gives 15.
15 is 2 times 7, plus 1: odd, as the proof says.
Right!
Let's look back.
First: write it with letters.
Then: put them in.
Then: work it out.
Then: show the form.
Last: check.