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.
Best Answer
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.