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How to prove that any integer $n$ which is not divisible by $2$ or $3$ is not divisible by $6$?

The point was to prove separately inverse, converse and contrapositive statements of the given statement: "for all integers $n$, if $n$ is divisible by $6$, then $n$ is divisible by $3$ and $n$ is divisible by $2$". I have the proof for converse and inverse similar to that given in comments. I have trouble only with the proof that integer not divisible by $2$ or $3$ is not divisible by $6$.

As I review my proof for inverse statement, I'm not sure of it as well. "For all integers $n$, if $n$ is not divisible by $6$, $n$ is not divisible by $3$ or $n$ is not divisible by $2$."

$n = 6x$ where $x$ is not an integer
$n = 2\cdot 3\cdot x$
$n/2 = 3x$ and $n/3 = 2x$ where $2x$ or $3x$ is not an integer,
so $n$ is not divisible by $2$ or $3$

marco trevi
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Victo
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    One line with contrapositive. What have you tried, and where are you having difficulty? Please [edit] your question to include something more than a problem statement; this is not a do-my-homework site. –  Oct 07 '18 at 22:11
  • By proving that if it is divisible by $6$ then it is divisible by both. – fleablood Oct 07 '18 at 22:35

2 Answers2

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If $6\mid n$, then $n=6\cdot k$ for some $k$. So $n=2\cdot 3\cdot k$. Thus $n$ is divisible by $2$ and $3$.

  • Perhaps it would be nice to give the asker some time to think about the question with some prodding, rather than just handing them a solution. –  Oct 07 '18 at 22:18
  • I have similar proof for converse statement: "for all integers n, if n is divisible by 3 and n is divisible by 2, then n is divisible by 6" – Victo Oct 08 '18 at 06:39
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The contra positive statement is that every multiple of $6$ is a multiple of $2$ and a multiple of $3$

Can you prove the contrapositive?

Mohammad Riazi-Kermani
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  • That I did, but the point was to prove separately inverse, converse and contrapositive statements. I have trouble only with the proof that integer not divisible by 2 or 3 is not divisible by 6. – Victo Oct 08 '18 at 06:26