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

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 $$n$$.

(i) necessary,

(ii) none,

(iii) sufficient,

(iv) sufficient,

(v) sufficient and necessary,

(vi) sufficient and necessary,

(vii) necessary.

• Looks right to me. Oct 30 '16 at 13:13

$$9$$ 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.
• "$6$ 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 $n$" Oct 30 '16 at 18:39