# ##n^2## is divided by 3, then n is also divided by 3.

## Homework Statement

Prove that if ##n^2## is divided by 3, then also n can also be divided by 3.

## The Attempt at a Solution

Is there anything wrong with this argument?

By contradiction, suppose ##3|n## and 3 doesn't divide ##n^2##.
Then ##3m = n##. Multiple both sides by ##n##
$$3(mn) = n^2$$
Thus, ##3|n^2## and we have reached a contradiction. Therefore, If ##3|n^2##, then ##3|n##.

## Answers and Replies

Nathanael
Homework Helper
Right. If n=3m then n2=9m2 which is always divisible by 3.

By contradiction, suppose 3|n3|n and 3 doesn't divide n2n^2.
This is not right. Your prove that the conjecture 3|n Λ not 3|n^2 produces a contradiction, but that proves that 3|n ⇒ 3|n^2.
You need to prove that 3|n^2 ⇒ 3|n

HallsofIvy
Science Advisor
Homework Helper
What you have proved is that "if n is divisible by 3 then $n^2$ is divisible by 3. That's the "wrong way around"!
To give a proof by contradiction, you need to start "suppose n is NOT divisible by 3", not that is divisible by 3.

Do you see that if n is not divisible by 3, it must be of the form "3k+ 1" or "3k+ 2" for some integer k?