Step by step
The question
Question: prove that the product of an even number and an odd number is even.
1 Write it with letters
x is even, so x = 2a, where a is a whole number.
That is what even means.
y is odd, so y = 2b + 1, where b is a whole number.
Careful!
An example shows one number.
The letters show every number at once.
2 Put them in
Now put them in: xy becomes (2a)(2b + 1).
3 Work it out
Every part times every part: 4ab + 2a.
4 Show the form
Every term has a 2 in it.
We take out 2: 2(2ab + a).
2ab + a is a whole number, because whole numbers added and multiplied stay whole.
2 times a whole number: so xy is even.
The square at the end means: the proof is done.
5 Check
An example: x = 2 and y = 5.
It gives 10.
10 is 2 times 5: even, 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.