Proving Divisibility of n by 6: A Simple Guide

[ToastMonger]

I need to prove that if n is a natural number and n^2 is divisible by 6, then n is divisible by 6.

I know that I knew how to do this at one point fairly recently, if you could refresh my memory I would greatly appreciate it.

 

[DeadWolfe]

Write n out in terms of it's prime factors, and it should be obvious.

 

[CRGreathouse]

Suppose n was not divisible by 2. Then n^2 could not be divisible by 6.
Suppose n was not divisible by 3. Then n^2 could not be divisible by 6.
So n is divisible by 2 and 3, hence by 6.

 

[murshid_islam]

we know that if p divides ab, then p divides a or p divides b.
now if 6 divides n.n, then 6 divides n.

 

[morphism]

Here's another way to look at it.

n(n+1)(2n+1)/6 is the sum of the first n squares, and thus must be an integer. Its numerator is 2n^3 + 3n^2 + n. So if n^2 is divisible by 6, n must be as well.

 

[Hurkyl]

we know that if p divides ab, then p divides a or p divides b.
now if 6 divides n.n, then 6 divides n.

6 isn't a prime number, so we cannot apply that theorem directly...

 

[Gokul43201]

Why are folks writing up complete solutions to what appears to be a textbook problem?

 

[murshid_islam]

we know that if p divides ab, then p divides a or p divides b.
now if 6 divides n.n, then 6 divides n.

6 isn't a prime number, so we cannot apply that theorem directly...

now if 6 divides n.n, then 3 divides n.n and 3 divides n.
again, if 6 divides n.n, then 2 divides n.n and 2 divides n.
so then 3 divides n and 2 divides n. therefore, 6 divides n.

 

[Gokul43201]

Great! That makes 3 complete solutions for the OP to copy down. Anyone else?

People, please read and follow the Posting Guidelines.

https://www.physicsforums.com/showthread.php?t=5374

Homework Help:
On posting questions: Any and all high school and undergraduate homework assignments or textbook style exercises for which you are seeking assistance are to be posted in our Science Education Zone. This should be done whether or not the problem is part of one's coursework. The reason for this is that the scientific and mathematical sections of Physics Forums are to be reserved for discussions and not academic assistance. Since graduate level assignments are meant to be more thought provoking (and hence more worthy of discussion), graduate level questions will be allowed in the relevant part of the main section of PF, provided that the graduate student attempts the problem and shows his work. NOTE: You MUST show that you have attempted to answer your question in order to receive help.

On helping with questions: Any and all assistance given to homework assignments or textbook style exercises should be given only after the questioner has shown some effort in solving the problem. If no attempt is made then the questioner should be asked to provide one before any assistance is given. Under no circumstances should complete solutions be provided to a questioner, whether or not an attempt has been made.

 

[fishingspree2]

Hello, my math skills are crap, but can't we simply do that?
If n is divisible by 6, then it can be writed as 6x where x is an integer.

$$\left(6x\right)^{2}=6^{2}x^{2}=36x^{2}=6\left(6x^{2}\right)$$

 

[dodo]

You could do that... if the intention was to show "if n is divisible by 6, then n^2 is too". But the original question was the other way around: "if true for n^2, then it's true for n".