How to determine sufficient and necessary conditions?

Question

Which of the following conditions are

a) necessary , b) sufficient for the natural number $n$ to be divisible by $6$?

(i) $n$ is divisible by $3$

(ii) $n$ is divisible by $9$

(iii) $n$ is divided by $12$

(iv) $n$ is even and divisible by $3$

(v) $n=132$

please tell me how you were able to solve it?

Many people here think that the use of allcaps is the same as shouting...please edit your post. As to your question...you must have some idea. What about the fifth one? $n=132$ . Is that sufficient? Is it necessary? — lulu, Sep 16 '18 at 12:58
See the post What is the difference between necessary and sufficient conditions ? — Mauro ALLEGRANZA, Sep 16 '18 at 13:01
"if $A$, then $B$" must be read : "$A$ is a sufficient condition for $B$" and "$B$ is a necessary condition for $A$". — Mauro ALLEGRANZA, Sep 16 '18 at 13:03
The first step in solving these problems should be to look up the definitions of "sufficient" and "necessary"1 — user247327, Sep 16 '18 at 13:06
And intuition must help : $3$ divides $15$ but $6$ does not. This means that to be divisible by $3$ is not sufficient in order to be divisible by $6$. But if a number is divisible by $6$ it is obviously divisible also by $3$. This means that divisibility by $3$ is necessary for the divisibility by $6$. — Mauro ALLEGRANZA, Sep 16 '18 at 13:14

score 0 · Answer 1 · answered Sep 16 '18 at 13:14

The first step in solving these problems should be to look up the definitions of "sufficient" and "necessary"! Condition "A" is "sufficient" for condition B if knowing that A is true immediately tells us that B is true. Knowing that "n is divisible by 3" does NOT immediately tell us that "n is divisible by 6" (9 is divisible by 3 but not by 6) so "divisible by 3" is NOT "sufficient" for "divisible by 6". Condition "A" is "necessary" for condition "B" if knowing that A is false immediately tells us that B is false. Knowing that n is NOT divisible by 3 does tell us immediately that n is not divisible by 6 so "divisible by 3" is a necessary condition for "division by 6"

How to determine sufficient and necessary conditions?

1 Answers1