I need help with the following problem:  Let $n$ and $m$ be positive integers.   Prove that  $$\frac{\gcd(n,m)}{n}{n \choose m}$$  is an integer.
Prove that for every $n\geq m \geq1$ natural numbers, the following number is an integer:  $${\gcd(n,m)\over n}\cdot{n\choose m}$$  Where $\gcd$ is the greatest common divisor.  I tried to make it simpler by cancelling the $n$ from the left side, and making it $(n-1)!$ on the right: $\gcd(n, m) \cdot \frac{(n-1)!}{m!(n-m)!}$, but can't really go further.   This was problem B-2 on the .
Boarding a train in India at another station I have booked a ticket on a train from Mumbai to Chennai and boarding at Pune. Since my schedules have changed, I will be in Mumbai on that day. Can I board the same train in Mumbai in my own ticket and claim the seat/berth? “Fire” a weapon before firearms existed? Did the verb “fire a weapon” exist before the actual introduction of firearms on battlefields?  More specifically, does it make sense for a creative work to have archers (or whatever ranged weaponry) be told to “fire!”, when the world they live in has not yet seen firearms?  It seems some kind of an anachronism to me, since before firearms, “fire” would never propel any projectile...  I've seen several movies do it; I can't remember them all, but for instance I verified it in The Lord of the Rings trilogy (left), Kingdom of Heaven (top right), and more recently Frozen (bottom right):    The first two do it several times (with Kingdom of Heaven also using it on ballistae, and The Return of the King even having Aragorn ask Legolas to ), whereas firearms never appear on the battlefields they feature (Kingdom of Heaven is set in the 12th century; as for The Lord of the Rings, apart from one occurrence of a witchcraft-ish bomb, it's archery and medieval siege engines all the way).  Frozen may be debatable, happening probably somewhere in the 18th or 19th century, but since we only see swords, spears and the like, it made me flinch to hear crossbowmen use “fire!”.  I've also noticed other movies avoiding this, using “loose!” instead, such as Troy (left) and Gladiator (right):    This makes me think it would be the right thing to say instead... although I'm still wondering why not use “shoot”, which seems simpler to me. JavaScript property access: dot notation vs. brackets? "I'm lovin' it" How normal-sounding is the slogan "I'm lovin' it" to native ears?  I know it sounded quite odd to me when I first heard it — and it still sometimes does —, but I can't even tell why. Sure, love is supposedly a stative verb, but it's being used in progressive aspect all the time without sounding weird at all (lots of songs come to mind, e.g. ). It's only this particular slogan that somehow doesn't quite work for me. I would expect that to be totally on purpose — it's advertising, after all —, but Wikipedia  any objections from native speakers (as it does with "" and ""). So I'm asking the native speakers of this community: does "I'm lovin' it" sound completely natural to you? Just a bit off? Completely weird? Why? Uploading video to youtube in right format I have a Sony CX115 and I have a little footage I'd like to put on youtube. I've downloaded Arista from the Ubuntu Software Center and then I've downloaded its Youtube HD plugin. I've converted my video and uploaded it to youtube... The result is a flashy video with a lot of grey in it:    What could I do to publish it in high quality so it'll be viewable? Who invented "dooblidoo"? The word  is used by several different youtube channels as a different word for the youtube description bar. I've seen it used by the  and by .  Who was first to use and invent the term? Travelling to Italy without my spouse who is a UK citizen Numbering equations If I am writing a LaTeX document on TexMaker and I want to number my equation but most probably I will be adding details later on and I do not want to fall into retyping it all over again. So if for example I have an equation and currently called equation (1.2) that I want to refer later as in the following sentence "As you can see in (1.2)" but then I had to edit my document in a way that it becomes (1.3) now instead of (1.2) what should I do in order to avoid editing large number of equations and my reference to them? I can't create an onCheckedChanged listener for my RadioGroup Sending multiple messages through only one SocketChannel Limit problem arctg (1 ^ infinty) Can anyone help me with this limit problem without L'Hopital rule and Taylor series? $$\lim_{x\rightarrow\ -\infty}\left(-2\frac{\arctan(x)}{\pi}\right)^x$$ Short story identification: society partitioned off by high wall Reading another question today reminded me of a short story I read in the early 90s (probably written in 60s or 70s though). A society is blocked off by a really high wall; the story follows a girl who tries to get over the wall. I think she manages it with a hot air balloon. When she reaches the other side she meets an older man who had managed to scale the wall in his youth - his solution had been to use a flock of birds to carry him.  UPDATE  I seem to recall that it splits off an elite group of people - getting over the wall is like a rite of passage. Most people don't even bother to get over it (or even be concerned by its existence at all), but a few try. There might be some magic involved (on the other / better side), or they might just be more advanced technologically.  FINAL UPDATE  As pointed out in the answer, and confirmed by me re-reading the story, I was mistaken in a few details. The main protagonist is not a girl, it's a boy called Porgie. He doesn't use a balloon, it's a glider he made, assisted with a broom. Magic is involved, but it's on the inside, with technology on the outside. Is it possible to use multiple desktop environments on same system? Why can current only flow in loops? In the following image:   Why can't current flow across the following wire? It's a simple question, but I've kind off always wondered.  Thanks! Windows 8 skype is terrible.  They've removed many buttons, most of the preferences, and have in general made it impossibly oversimplified and difficult to use.  Old Skype was fine when I had windows 7.  Is there any way to get it back on Windows 8? In Python, what's the difference between 'except Exception as e' and 'except Exception, e' Why doesn't Volumetric follow the shape of Mesh in Eevee? So I was in the middle of making a ball-shaped smoke, but the volumetric effect didn't work following the basic shape. It remains cube in Eevee Blender 2.8; while in Cycle it works, but in Cycle it takes thousands of years for Render. The ideal generated by $2$ and $ x$ in $\mathbb{Z}[x]$ Storage for Eldritch Horror Is there an optimal strategy for storing Eldritch Horror cards and tokens so that game play is facilitated.   I am not looking for answers that including "zip bags" and other things to keep the game element safer while is storage. Although, a strategy that works well for both cases will clearly be superior to one that only works for game play.  I am not limiting this to commercial solutions. I am quiet happy to build your custom made solution if you release the specification/plans of course!  Although, it it requires a 3D printer, more power to you. Is the cardinality of the Cartesian product of two equinumerous infinite sets the same as the cardinality of any one of the sets? I couldn't find this explicitly stated in any handout or text. This certainly seems to be true from the examples I have seen:  The Cartesian product of two infinitely countable sets is again infinitely countable. The Cartesian product of two sets with cardinality of continuum again has cardinality of continuum.