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#Pedagogy, #research etc:
I provided a solution of a math olympiad problem yesterday as a comment (paraphrased by Yaashaa P. Golovanov as "the exact solution").
If I ever have to teach a number theory class again (those days are over – I did teach such a course and a few of my connections here were in that class, including a then high school kid Ishan Paranjpe, MD), I would design the following homework problem (not for an in-class exam):
For a given positive integer m, (a) show that there are infinitely many positive integers n satisfying the following property:
The decimal representation of 5^n ends with the same as that of 5^m.
(b) Also provide a complete characterization of all such positive integers n.
Part a is easy for anyone who can solve the analogous problem given in the shared post here.
Part b does require elementary group theory (Lagrange’s theorem which was discussed in an older post of Yaasha wherein I rendered my 2-cents of wisdom)
Hint for (b) (which also serves as hint for (a)) All n of the form m+k*d, where k =0,1,2,3,… and d is the smallest positive integer that satisfies 5^d = 1 (mod 2^m);
Those who are familiar with elementary group theory would recognize: d = order of the element 5 in the multiplicative group G of units modulo 2^m. Easy to see that this group G consists precisely of all the odd integers between 0 and 2^m, hence G has order 2^(m-1).
A toy example when m = 3. The solution for n from the original post for the case m = 3 is 3+\phi(8) = 3 + 4 = 7, but a smaller value of n would also suffice noting d = 2 since 5^d=5^2=1 (mod 8), so n = 3+2 = 5 works too.
Pedagogical advice from an old dude (i.e. my humble self) to the young and aspiring researchers/scientists: as you from above (see the BIG picture), original problem asked for only one value of n. Then I asked for more such n. How many more such? The more the merrier. Then we ask, can we find ALL
such n (so-called characterization problem)? As a by-product, we asked: is the n we found the smallest, can we find even a smaller n? and then onto their characterization and so on. This is "greedy, hungry, thirsty" in research. Beat the problem to death!!
What is the next natural question? Instead of base 5, ask the same question with OTHER bases: for any fixed k, k^n and k^m? The answer is sort of trivial and mundane generalization of what we have outlined above solves the problem completely for ALL m and ALL bases k, also providing complete characterization of all such n in this generality.
The word “characterization” means: prove that the values you claim as n work and nothing but those values would work (so-called “if and only if” condition).
This is a baby tutorial on how to carry out a mathematical research; I suppose this philosophy carries over to most other disciplines.
Thanks to Yaasha for giving me an opportunity to pen this pseudo-blog.
#mathematics#stemManish K. GuptaNawal Kishor MishraAlexander GroegerVINISHA UMASHANKAR
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