Is 3 Divisible by n if 3 is Divisible by n^2?

  • Context: Undergrad 
  • Thread starter Thread starter cubicmonkey
  • Start date Start date
Click For Summary

Discussion Overview

The discussion revolves around the question of whether 3 being divisible by \( n^2 \) implies that 3 is also divisible by \( n \), where \( n \) is an integer. This inquiry is tied to proving the irrationality of the square root of three and involves mathematical reasoning and proof techniques.

Discussion Character

  • Mathematical reasoning
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant expresses confusion about the proof and seeks assistance in understanding the implication that \( 3|n^2 \) leads to \( 3|n \).
  • Another participant notes that since 3 is a prime number, if 3 is not a prime factor of \( n \), it cannot be a factor of \( n^2 \).
  • A different participant acknowledges the contrapositive proof suggested but requests further elaboration on the proof itself.
  • A participant reiterates their confusion and presents a modular arithmetic argument, suggesting that if \( n \equiv a \pmod{3} \) and \( a \neq 0 \), then \( n^2 \equiv a^2 \pmod{3} \) leads to a contradiction, implying \( 3 \nmid n^2 \).
  • Another participant proposes that the Fundamental Theorem of Arithmetic, which discusses the uniqueness of prime factorizations, might be relevant to the discussion.

Areas of Agreement / Disagreement

The discussion contains multiple competing views and remains unresolved, with participants expressing differing levels of understanding and approaches to the proof.

Contextual Notes

Participants have not reached a consensus on the proof's validity, and there are unresolved assumptions regarding the implications of divisibility and the application of modular arithmetic.

cubicmonkey
Messages
6
Reaction score
0
I'm completely lost on this one. I need this to be able to solve that the square root of three is irrational. So it's a proof within a proof, but I like this way best. Please help me out.
This is what I need to know how to prove.

3|n^2 implies 3|n where n is some integer.
 
Physics news on Phys.org
3 is a prime number. If 3 were not a prime factor of n, it could not be a factor of n2.
 
I think I understand what you're saying and that looks like a contrapositive proof, but you don't actually prove it. Could you elaborate?
 
cubicmonkey said:
I'm completely lost on this one. I need this to be able to solve that the square root of three is irrational. So it's a proof within a proof, but I like this way best. Please help me out.
This is what I need to know how to prove.

3|n^2 implies 3|n where n is some integer.

Suppose :

n \equiv a \pmod 3 ~\text{and}~ a \neq 0
then :
n^2 \equiv a^2 \pmod 3 ~\text{and}~ a^2 \neq 0
hence :
3 \nmid n^2
contradiction .

Q.E.D.
 
I think the Fundamental Theorem of Arithmetic (uniqueness up to order of prime factorizations) might help?
 

Similar threads

Undergrad ##S_{n}(\alpha)=\sum_{k=1}^{n} (-1)^{\lfloor{k\alpha\rfloor}}##
  • · Replies 2 ·
Replies
2
Views
2K
Undergrad N-Dimensional Real Division Algebras
  • · Replies 6 ·
Replies
6
Views
2K
Prove: If the integer ## a ## is not divisible by ## 2 ## or ## 3 ##....
  • · Replies 9 ·
Replies
9
Views
2K
Undergrad I think I've hit upon a very easy proof of impossibility of quintic
  • · Replies 2 ·
Replies
2
Views
2K
Undergrad Trouble understanding an online solution to an exercise in Dummit & Foote
  • · Replies 21 ·
Replies
21
Views
2K
Undergrad How to show ##p(x)=g(x)x\pm 1\in\Bbb{Q}[x]## is irreducible in ##\Bbb{Q}_{\Bbb{Z}}[x]##?
  • · Replies 48 ·
2
Replies
48
Views
5K
Undergrad Matrix rank in terms of determinant polynomial
  • · Replies 14 ·
Replies
14
Views
4K
Proof That an Integer is Divisible by 2
  • · Replies 3 ·
Replies
3
Views
2K
Undergrad Base Pi Integers: Isomorphism with Rationals?
  • · Replies 33 ·
2
Replies
33
Views
4K
High School Probability that 2 distinct integers are divisible by the same number
  • · Replies 9 ·
Replies
9
Views
2K