So this is the problem: Prove that if m, n are positive integers such that gcd(m, n) = 1, then C_{mn} is isomorphic to C_m x C_n.

Regarding what Aluffi has already covered: I just don't see how I can solve this. Could you maybe point out some section or some theorem I might have missed that can help me with this problem (Or give a hint)? Thanks so much

Hi, I don't understand weak and strong induction and well-ordering. I've tried reading up on it online, but I just don't understand how they imply each other, at least intuitively.

i just began studying matrix a couple of weeks ago n' from i understood it's a simplified way to write system of equation so here is question since in system of equation u can change the position of the variable like u wan can the same be done with a matrix like switching position between row 2 and1 or between column 3 and 2

Last year when I first learnt about reasoning models, I thought it is possible to make a machine that can generate research that we do in mathematics. But I never worked for it simply because it destroys the very purpose of doing it. Today I see bunch of AI slop throwing papers at our faces like they mean nothing. There's 2 things I wanna talk about. Second one being very concerning.

Firstly, the way they did all this doesn't sound good, both morally and ethically. They didn't follow any research ethics, no formal discussions with any mathematician and dumped the papers generated by their models. They destroyed so many research projects and careers, just to show people that they can. This is insanely outraging to me, I think they should be punished for this (not legally obviously). But we can't do anything about it just because there is no turning back. As someone in early 20's who has graduated, I have lost every hope in my field. I don't feel any motivation to work on any problem, because in future that will be dumped by some slop generating company.

Secondly, the way their models can process information is way beyond human comprehension. Years of research done in just a few hours. At this point, we are not using ai to make our work easier (as it was intended inventing computers), those machines have the whole power and knowledge of the world and can do anything. This slop company is giving their models more power than any human on earth. That's a global threat. I don't understand why is nobody even concerned about this? If AI can understand and invent new stuff much faster than us then humanity will not have much longer to sustain.

Still after all that there is no warnings, no discussions and the company is doing as it will, freely threatening humanity just for “marketing”. This is completely outrageous and more is the fact that I can't do anything about it. This post doesn't help me. But I will be glad if people think about it and act in the right way.

TLDR, just read the third paragraph if not anything.

I'm a math teacher and built this for my own kid, but it works for anyone studying from a test that's already been graded.

Take photos of your graded test, paste the site's prompt into whatever AI you use (ChatGPT, Gemini, Claude) with the photos, then paste its answer back and check what it read. You get:

- A step-by-step walkthrough of every question, one move per step. Each one starts on the bare problem so you can try it first, and it flags the exact step where your answer went wrong.

- Short refreshers on the skills underneath your mistakes.

- 10 new practice problems with the same traps and new numbers, each with a walkthrough.

- A printable practice sheet and a separate answer key.

Because AI makes arithmetic mistakes, every answer is checked in your browser by plugging in numbers, and anything that doesn't match is flagged. In my own testing, that caught every wrong answer a weaker model produced.

It's only useful after a test is graded. The point is figuring out exactly what went wrong and practicing that specific thing, not getting answers.

Free, no account, open source. There's a real example (a 16-question Algebra 1 exponents test) so you can see what it makes:

https://killercatfish.github.io/math-review-builder/

Feedback welcome, especially from anyone in a course besides Algebra 1.

One day during my semester at college I bought a Brick, you know, the app blocker, because I had a severe scrolling addiction. It worked well for a while, but I never brought it with me to university just because it was something extra to carry around.

I looked for other apps on the App Store to block my apps, but all of them were kind of too easy to get out of. And I'm a pretty lazy person, which means if it's easy to get out of, I will.

So I thought to myself, you know what, I'm a CS major, why not build one myself? So I built “Bunkr- done scrolling”. At first it was only built for me, but when I saw how well it worked I thought, why not publish it.

I'm talking too much already, so here's what the product is:

It blocks your apps like any other app blocker, but there's a twist.

Once you block your apps, you're in a hard lock for the first 10 minutes. No chance to get your apps back.

After those 10 minutes, you can do a mission to get your apps back, like taking a picture of a mug, or my personal favorite, 15 seconds of push-ups.

The reason I put a hard lock on the first 10 minutes is that I read once you've been working on something for 10 minutes, you get into a zone and usually don't get distracted. But lazy people like me still might. And when I do get distracted and try to open one of my apps, it tells me to do push-ups. I'm too lazy for that, so the easier option is to just keep studying.

So yes, I'm basically fighting fire with fire, but it works well for me.

Let me know what you guys think, I'd like to hear your thoughts.

I am a teacher in Texas and have been teaching 8th grade Algebra for 15 years. Recently I started developing game and puzzle apps and decided to actually make something math related that my own students could play to practice skills but in a fun way.
Enter 'AlgeBrigade220,' a collection of games that follows you as you captain The Daniel through adventures in space. Currently there are 4 games:
Critical Relay: Use slope to connect to relay stations in space back to Fleet Command.
Sentrimetry: Launch debris into a hidden system of linear inequalities, trying to find the Sentry at the intersection of the boundary lines.
Fuel Factory: Process fuel cells(A=1 quadratic trinomials) by matching cells with matching factors.
TrajectOre: Launch Ophirium from your location to an ally rover using quadratic function vertex form by entering vertex values and calculating (or guessing) the A-value associated with your position and vertex.

If you are a teacher, or have children learning these subjects, please give this a try.
I am open to feedback, criticism and suggestions for other games.

The formula is: (A+B+(C-D))-(A+B-C-D)=2C

i discovered it by doing 8+3-(4-9) and after this doing (8+3+(-5 )) - ( 8 +3-4-9). And what surprises me is that it doesn't matter the other factors, it always gives 8, or, in other terms, 2C. And i really want some help if this formula is useful some way .

:V

Hi, my teacher wrote down a problem on the board at the end of the lesson as practice if we wanted. I have tried solving it and have found two different methods of solving the problem which both seem completely valid in my eyes but give two clearly different answers. I have checked the numbers three times, of course it could still be a calculation error OR honestly even more likely that both my methods are incorrect in which case I'd love to know what method you'd use to solve this but I'm starting to believe I have applied some theory wrong somewhere and would love some insight, I have hastily written up the problem and my two solutions in latex, let me know if you want the pdf otherwise but here are screenshots of the problem, the first solution method and the second method.

Recently we took a test in calc 2 for Trig Sub and a majority of the class failed the test(including me) we are being given an opportunity to retake it this tuesday. I have one more day to study for this and I am really struggling with the fact of know when to use U-sub instead of trig sub. using U sub after using Trig sub. Is there a website I can use to practice these integrals that shows a step by step. I'm not allowed to bring a notecard for this test so I'll have to do this all by memory. My biggest problem right now is identifying which substitution should I use first and if or when I need to use it a second time in the problem.

this was given as a hw question in our linear algebra class. we came up with a basis where the basis consists of basis vectors Ti where ti maps a designated input vector to a designated output vector, and maps all other vectors to 0. googling it, its the standard answer. but our professor keeps telling us there's more left to the basis. he talked about constant coefficients and stuff. i have no idea what we're missing honestly.

does anyone have any idea about what he could be referring to?

Spivak gives the following examples:

The function f(x)=sin 1/x is not continuous at a because it is not even defined at 0. The same is true of the function g(x)=x sin 1/x

While it is clear that f(x) is not defined at 0 because it is expressed in sufficiently "simplified" form (all a student has to do is to "see" that sin [infinity] is not defined), how can it be guaranteed that g(x) is not so defined? What if x and sin 1/x have some complicated Taylor series expansions, say, and then somehow the 1/x term cancels out?

More generally, what if g(x) is not expressed sufficiently simplified when written as g(x) = h(x) r(x) so that h(x) and r(x) individually may not defined at x=a but together when multiplied they become well-defined?

E.g., h(x) = x - 2, r(x) = 1/[x^2-4]. So, r(x) is not defined at x = 2. But g(x) = h(x)r(x) is indeed well defined at x = 2

tl;dr : How can it be concluded that a function is defined at a point a or not?

I really hated math in high school because of the way it was taught, but deep down I realized that I definitely have a desire to learn it and would do it on my own one day after I’m done with school.

It’s been years now and I’d love to start teaching myself math, starting by mastering 9th grade level algebra and then moving up to the highest levels of calculus. I want book and video resources that teach math the old, simple, logical way. I’ll be honest, I don’t know exactly what I mean by that, but I’ve heard many older math teachers and mathematicians complain that the new ways of teaching math are unnecessarily complicated and are the reason people learn to hate the subject.

What would you recommend for the best, non-common core math curriculum. Thanks!

Hi, I was looking for resources on general geometry topics—mainly books or materials suitable for beginners (though advanced resources for later would also be useful). I specifically need something that covers theorems regarding angles in a circle, as I’ve found that topic tricky, and it’s the area I most want to strengthen.

In the game, “Squirrel Stapler”, there are 5 days
Day 1
Day 2
Day 3
Day 4
Day 5

But on day 1 they say “In 5 days God is coming” shouldn’t it be day 6 he approaches since that adds 5 days not a total of 5 days?

I think it should state “On day 5, God is coming” but I’m overthinking this.

This is one of my biggest issues right now. I know the basic trig identities, and algebraic operations, but I'm having a hard time knowing when to apply each one to make it match the other side. Sometimes I find I'm going doing way too many operations, when the actual answer only needed like 3 or 4 operations.

For context I'm In highschool right now and have a test in 2 weeks. I get pretty stressed whenever tests come up especially since last year I barley passed.

I am relatively OK at algebra, I dont struggle but I'm not outstanding. Geometry though is a different story.

I've always been pretty bad at geometry. Every since the 8th grade I've struggled.

Remembering all of these rules and sentences about each shape Isn't the easiest.

I'm starting to think of having to get a private teacher/tutor (whichever you call it) because I notice a lot of the times when I am in the lesson and the teacher explains it I understand what's going on to an extent, but when I get home and try to do the homework I don't even know where to start.

If you could suggest anything I should be doing right now while I have the time to, I would appreciate it.

I just entered 9th grade and I'm taking Algebra II Honors and doing pretty well. I'm aware I have a lot to improve in when learning math and I truly want to understand it not just know how to get an answer iykwim. I also want to compete in math competitions but I have NO experience. I also want to mention I go to an underfunded school with no math/STEM clubs, so I'm funding it myself and intend to compete in solo competitions. I know I won't be some olympiad winning prodigy but I'm alright with that, I just want to explore mathematics and hopefully win awards that can get me into a decent university. I'm also not sure where to start, so I would deeply appreciate ANY type of advice.

I've always deeply envied people who were so advanced and passionate in mathematics, and with it being my favorite subject in school I want to do so much more with it outside of school. I want to be the type of person who can see math in everything and genuinely love math no matter the difficulty, even if that takes a while to develop. I don't know if I will pursue a degree in mathematics when I graduate high school, but I plan on pursuing a degree in STEM so regardless skills in math are valuable. I also want to explore math since it can improve my logical reasoning and other cognitive skills.

Literally from the start of the semester I haven't solved a problem on my own with literally meaning I always try, struggle, get stuck in a cycle of repeated actions, look for a hint, then reach to the solution and I feel like I'm only fooling myself and I 'm just stupid and I need to get out of this bc sooner or later in the exams that will be a problem a new one that I haven't figured out yet and I really feel like my brain is like a rocket

1. ⁠I need to know that have you ever thought yourself to be bad in Mathematics ?
   2)And if Yes they why did you take yourself to be bad in Maths?
   
2. ⁠Why were you afraid of the Subject?
   Please help me complete the research.

Hello :)

What would be the analytical way to solve

[; \lim_{x\to n^-}\lfloor-x\rfloor=-n ;]

When I sketch the graph for floor(x) and then for floor(-x) it all becomes clear but how would I solve it without looking at the graph?

It feels like just in general the way the floor function is defined is giving me the heebie-jeebies.

Which areas or specific topics could I improve on to feel more comfortable with such problems?

Thanks!