Simple Proof

  • Thread starter sapiental
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  • #1
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Homework Statement



Prove the following: If the integer n is divisible by 3 then n^3 is divisible by 3.

Homework Equations



Direct Proof

The Attempt at a Solution



n = 3m

n^2 = 9m^2

n^2 = 3(3m^2)

I think the proof is done at this point because the 3 factors out but I also did this:

n^2 = 9m^2

(n^2)/9 = m^2

(n/3)(n/3) = (m)(m)

which also implies n is divisible by 3 since integer x integer = integer

My professor is kind of harsh on proofs so Im not sure if there are intermediate steps I'm missing. Thanks!
 

Answers and Replies

  • #2
Gib Z
Homework Helper
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If 3 is a factor of n, it must also be a factor of any natural power of n. Do you mean n^3 or n^2? Your working looks like n^2. Anyway, the first way you did it is correct and satisfactory though if your teacher is really harsh, you may want to go like:

[tex]n= 3m[/tex] for some natural value of m, because n is divisible by 3 (Data).
Squaring both sides, [tex]n^2 = 9m^2 = 3 ( 3m^2)[/tex]. Since m is natural, 3m^2 must also be natural, and hence 3 is a factor of n^2 as well.
 

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