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Question: Are my answers correct? I am particularly concerned with (ii) and (vii).

Which of the following conditions are necessary for the positive integer $n$ to be divisible by 6? Which of them are sufficient?

(i) 3 divides $n$.

(ii) 9 divides $n$.

(iii) 12 divides $n$.

(iv) $n =$ 12.

(v) 6 divides $n^2$.

(vi) 2 divides $n$ and 3 divides $n$.

(vii) 2 divides $n$ or 3 divides $n$.

Answers:

(i) necessary,

(ii) none,

(iii) sufficient,

(iv) sufficient,

(v) sufficient and necessary,

(vi) sufficient and necessary,

(vii) necessary.

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    $\begingroup$ Looks right to me. $\endgroup$
    – Camille
    Oct 30 '16 at 13:13
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I think that your answer to numbers II and VII are correct.

$9$ is not a factor of $6$ so is not necessary.

Since $n$ might be divisible by $2$ but may not be divisible by $3$ (e.g. $4$) and since $n$ might be divisible by $3$ but may not be divisible by $2$ (e.g. $9$) VII is insufficient but must be necessary.

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  • $\begingroup$ So you are saying (vii) is neither sufficient, nor necessary? $\endgroup$
    – user335936
    Oct 30 '16 at 16:50
  • $\begingroup$ iit may be necessary $\endgroup$
    – Maharrnab
    Oct 30 '16 at 16:52
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    $\begingroup$ "$6$ divides $n$" $\implies$ "$2$ divides $n$ or $3$ divides $n$" - therefore, condition (vii) is necessary. However, using an exclusive-or would make (vii) into a sufficient condition for the opposite statement, "$6$ does not divide $n$" $\endgroup$
    – Joffan
    Oct 30 '16 at 18:39
  • $\begingroup$ TRUE on ur part $\endgroup$
    – Maharrnab
    Oct 30 '16 at 18:45

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