Step by step
The question
Question: prove that if n is even, then n² − 4n is a multiple of 4.
1 Write it with letters
n is even, so n = 2a, where a is a whole number.
That is what even means.
Careful!
An example shows one number.
The letters show every number at once.
2 Put them in
Now put them in: n² − 4n becomes (2a)² − 4(2a).
3 Work it out
(2a)² means the bracket times itself.
Every part times every part: 4a² − 8a.
4 Show the form
Every term has a 4 in it.
We take out 4: 4(a² − 2a).
a² − 2a is a whole number, because whole numbers added and multiplied stay whole.
4 times a whole number: so n² − 4n is a multiple of 4.
The square at the end means: the proof is done.
5 Check
An example: n = 2.
It gives -4.
-4 is 4 times -1: a multiple of 4, 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.