Step by step
The question
Question: prove that if n is a multiple of 3, then n² is a multiple of 9.
1 Write it with letters
n is a multiple of 3, so n = 3a, where a is a whole number.
That is what a multiple of 3 means.
Careful!
An example shows one number.
The letters show every number at once.
2 Put them in
Now put them in: n² becomes (3a)².
3 Work it out
(3a)² means the bracket times itself.
Every part times every part: 9a².
4 Show the form
Every term has a 9 in it.
We take out 9: 9a².
a² is a whole number, because whole numbers added and multiplied stay whole.
9 times a whole number: so n² is a multiple of 9.
The square at the end means: the proof is done.
5 Check
An example: n = 3.
It gives 9.
9 is 9 times 1: a multiple of 9, 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.