There's quite a few questions around (such as Show that 13 divides $2^{70}+3^{70}$) which ask a homework question which could correctly be answered in numerous ways. For the sake of discussion, I'd like to put them into two categories (of course, in reality, answers are not going to fit exactly in either category):
- ad hoc: where the techniques used in answering the question are likely to be unhelpful in resolving similar questions. These answers tend to be particularly slick and creative.
- pedagogical: utilising the techniques that one suspects the author of the homework question intended, written at a level chosen based on the original question. These answers tend to be unremarkable, but the techniques involved are likely to be useful to the student for subsequent questions.
To illustrate let's contrast two answers to the above referenced questions. Zarrax's answer would fit in the ad hoc category above:
...if $n$ is odd the polynomial $x + y$ divides $x^n + y^n$. So letting $x = 2^2, y = 3^2,$ and $n = 35$ you get that $13 = 2^2 + 3^2$ divides $2^{70} + 3^{70}$.
The issue I would like to raise here is that if the teacher of this subject changes the question slightly on e.g. an exam, then understanding this answer is likely not to be helpful. E.g. imagine if the teacher asked: "Prove that $2^{70}+3^{71}$ is not divisible by 13."
[To be fair to Zarrax, that answer is more of a comment-in-an-answer, and seems to be intended for the benefit of the whole community, rather than just the OP. (and it is a nice proof)]
Arturo Magidin's answer fits into the pedagogical category:
Compute $2^{70}$ and $3^{70}$ modulo $13$ separately (e.g., using Fermat's Little Theorem). If $2^{70}\equiv a\pmod{13}$ and $3^{70}\equiv b\pmod{13}$, then what is $2^{70}+3^{70}$ congruent to modulo 13?
Clearly this answer has the student as the primary focus. If the teacher then asks on an exam "Prove that $2^{70}+3^{71}$ is not divisible by 13" the student, after understanding Arturo Magidin's answer, should be able to do so.
I'm concerned that the focus of the community is too heavily on the side of the slick ad hoc answers.
- It might not be clear to the student which answers are intended for the overall community and which are intended primarily for the OP's benefit.
- There is a possibility that slick answers could inadvertently overwhelm a struggling student -- "I would never have thought of that answer; perhaps I'm not cut out for mathematics".
So with that I pass the discussion baton to the community:
Question: Is giving ad hoc answers to homework questions an issue? If so, does it need addressing?
I'd especially like to see feedback from students who utilise the forum to assist with their studies.