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

  • Thread starter cubicmonkey
  • Start date
In summary, the conversation discusses a proof within a proof regarding the irrationality of the square root of three. The focus is on proving that if 3 is a factor of n squared, then it must also be a factor of n. The conversation touches on the use of a contrapositive proof and the potential application of the Fundamental Theorem of Arithmetic.
  • #1
cubicmonkey
6
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
  • #2
3 is a prime number. If 3 were not a prime factor of n, it could not be a factor of n2.
 
  • #3
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?
 
  • #4
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 :

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

Q.E.D.
 
  • #5
I think the Fundamental Theorem of Arithmetic (uniqueness up to order of prime factorizations) might help?
 

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

1. What does "3|n^2" mean?

"3|n^2" means that 3 is a divisor of n^2, or in other words, n^2 is a multiple of 3.

2. How does "3|n^2" imply "3|n"?

If 3 is a divisor of n^2, then n^2 can be written as 3k for some integer k. This means that n^2 is a multiple of 3, and therefore n must also be a multiple of 3 in order for n^2 to be a multiple of 3. Hence, "3|n^2" implies "3|n".

3. Can you give an example of a number that satisfies "3|n^2" but not "3|n"?

Yes, for example, n = 6. 3 is a divisor of 6^2 (36), but 3 is not a divisor of 6.

4. How is this statement related to divisibility rules for 3?

This statement is essentially a divisibility rule for 3. It states that if a number is divisible by 3, then its square is also divisible by 3.

5. Is "3|n^2" a sufficient condition for "3|n"?

No, "3|n^2" is not a sufficient condition for "3|n". It only tells us that if "3|n^2" is true, then "3|n" must also be true. However, there may be other factors that contribute to "3|n" being true, such as n itself being a multiple of 3.

Similar threads

Back
Top