I'm trying to solve the following excercise (Houston: How to Think Like a
Mathematician; Exercise 27.23):
Consider the following statements
- (a) $n$ is divisible by 3,
- (b) $n$ is divisible by 9,
- (c) $n$ is divisible by 12,
- (d) $n = 24$,
- (e) $n^2$ is divisible by 3,
- (f) $n$ is even and divisible by 3.
Which conditions are necessary for the natural number n to be divisible by 6? Which are sufficient? Which are necessary and sufficient?
My thoughts about this:
- a) necessary
- b) nothing
- c) sufficient
- d) both
- e) necessary
- f) both
I'm not sure if my solutions are right. Perhaps someone can help me here?
Thanks a lot in advance.
Best Answer
All the answers are correct except for (d). Certainly the condition $n=24$ is sufficient for divisibility by $6$. But it is not necessary, since $18$ is divisible by $6$ but is not equal to $24$.
If this comes from a "short answer" or multiple choice test, the answers you gave are adequate, but some should be somewhat longer. If we want to be very brief, "both" has clear meaning in our context. The word "nothing" used in (b) is less clear; if one wants to be brief "neither" would be better.
However, in (c), writing down "sufficient" is not $\dots$ sufficient. After all, sufficient does not rule out necessary, so one should write for (c) something like "sufficient, but not necessary." Similar remarks apply to (a) and (e).
In a serious course, justifications would be expected. It is useful, even when "self-learning," to write down full details for a reasonable proportion of the problems that one solves.