Dividing $61!$ by $3$ as we can. Find maximum possible $n$ in the equation $61!=3^n\cdot m$.   Some textbooks gives a solution to this question like this: \begin{align*} 61&amp;=\textbf{20}\cdot 3+1 \\ 20&amp;=\textbf{6}\cdot 3+2 \\ 6&amp;=\textbf{2}\cdot3+0 \end{align*} $n_{max} = 20+6+2$. But this comes me unintuitive. And seems like it's working. Why this solution is true. Can you give an explanation?  I don't think the answer in  gives an understanding directly to solution i wrote since the algorithm continues dividing with the last full part.
Highest power of a prime $p$ dividing $N!$ How does one find the highest power of a prime $p$ that divides $N!$ and other related products?  Related question: How many zeros are there at the end of $N!$?    This is being done to reduce abstract duplicates. See   and  for more details.
Why must local variables have initial values? Points of a Measure Zero Sets Covered by Intervals Infinitely Many Times Given a measure zero set $E$, by definition we have forall $\varepsilon &gt; 0$, a covering of $E$ by intervals whose lengths sum to $&lt; \varepsilon$.  I want to prove that we can cover $E$ in intervals such that the sum of the lengths the intervals is $&lt; 1$ and each point in $E$ is contained in infinitely many of the intervals.  Do you know how to prove this? Thank you swap x and y axis on data generated plot Does a photon in vacuum have a rest frame? Does each imply the other? If not, does one imply the other? Why/why not?  This issue came up in response to a comment on an answer I posted .  Although google searching the relevant terms didn't produce anything that seemed particularly useful, I did notice  on the math stackexchange. However, I thought that this question was appropriate for this site too.  EDIT after reading the comments  Relative to the math.stackexchange answer I was after something more in depth, covering some of the issues dealt with in . Also, as I see it the math.stackexchange question shows that consistency doesn't imply asymptotically unbiasedness but doesn't explain much if anything about why. The OP there also takes for granted that asymptotic unbiasedness doesn't imply consistency, and thus the sole answerer so far doesn't address why this is. How to change symbol for footnote in minipage \renewcommand method does not work  \documentclass{article} \renewcommand{\thefootnote}{\arabic{footnote}} \begin{document}  \begin{minipage}{1.3\textwidth}  test this\footnote{abc} \end{minipage}  \end{document}   In the following example the footnote number is incorrect. Why?  \documentclass{article} \renewcommand{\thempfootnote}{\arabic{mpfootnote}} \begin{document} \begin{minipage}{.5\textwidth} test this\footnotemark \footnotetext{abc} \end{minipage} \end{document} Run Unity on external monitor with Bumblebee MLE for Uniform $(0,\theta)$ I am a bit confused about the derivation of MLE of Uniform$(0,\theta)$.  I understand that $L(\theta)={\theta}^{-n}$ is a decreasing function and to find the MLE we want to maximize the likelihood function.  What is confusing me is that if a function is decreasing, then wouldn't the function be maximized at the smallest input rather than the largest?  Thank you in advance for your help. What's the best way to develop old, exposed film? A friend of mine just found 3 rolls of exposed 135 (i.e. 35mm) film. They are at least 10 years old. He'd like to know if it is even worth taking them to the local hour photo lab but also if he should give them any special instructions. static variables in an inlined function I have a function that is declared and defined in a header file. This is a problem all by itself. When that function is not inlined, every translation unit that uses that header gets a copy of the function, and when they are linked together there are duplicated. I "fixed" that by making the function inline, but I'm afraid that this is a fragile solution because as far as I know, the compiler doesn't guarantee inlining, even when you specify the "inline" keyword. If this is not true, please correct me.  Anyways, the real question is, what happens to static variables inside this function? How many copies do I end up with? Joining lots of small polygons to form larger polygon using PostGIS? Axis with trigonometric labels in PGFPlots How do I make the axis labels use multiples of \pi in PGFPlots? How to install CUDA 8.0 on Ubuntu 16.04 with Nvidia GeForce GTX 1080 I am attempting to install CUDA 8.0 within Ubuntu 16.04 for the Nvidia GeForce GTX 1080. I had CUDA 7.5 installed, but it did not appear to be compatible with the GTX 1080. I downloaded the .run installer from Nvidia's developer page, but the installation fails and outputs the following to a nvidia-installer.log     An NVIDIA kernel module 'nvidia-drm' appears to already be loaded in     your kernel.  This may be because it is in use (for example, by an X     server, a CUDA program, or the NVIDIA Persistence Daemon), but this     may also happen if your kernel was configured without support for     module unloading.  Please be sure to exit any programs that may be     using the GPU(s) before attempting to upgrade your driver.  If no     GPU-based programs are running, you know that your kernel supports     module unloading, and you still receive this message, then an error     may have occured that has corrupted an NVIDIA kernel module's usage     count, for which the simplest remedy is to reboot your computer.   I have rebooted the computer, but that did not resolve the problem. How do I unload the 'nvidia-drm' module so that I can install the 8.0 CUDA? What is the moment of a joint random variable? Simple question, yet surprisingly difficult to find an answer online.  I know that for a RV $X$, we define the kth moment as $$\int X^k \ d P = \int x^k f(x) \ dx$$ where the equality follows if $p = f \cdot m$, for a density $f$ and Lebesgue measure $m$.  So, what is the kth moment of, say, $(X,Y)$? $\int (X,Y) \ d P$ doesn't really seem like the answer to me.... Suppose that $X$ is a topological space equipped with a delta-complex structure. Prove that $X$ is Hausdorff.  I can actually "see" why it should be Hausdorff after the hint from Hatcher asks to "think of a delta-complex X as a quotient space of a collection of disjoint simplices".  I would appreciate if somebody can write out a full formal argument showing this proof. I'm trying to start from the definition of Hausdorff but I don't know how to carry on. How do I learn to become a good GM? I have played and tried to GM games since I was in elementary school, beginning with games like Shadowrun, Earthdawn, and Champions. Recently, I've gotten dack into gaming with a variety of other games (newer Shadowrun and Earthdawn, plus TOON! and Outbreak:Undead).  I've found examples of actual play and advice on how to actually GM a game in games to be lacking. I can only help but feel I am not alone as someone who isn't already a LITERAL dungeon MASTER.  I fully realize that there are an unlimited number of GM styles, personalities, etc. I do not think I am a very good GM though. No one else wants to be the GM, and few care enough to learn the complex rules that I often have to assist with and explain AS WE PLAY. Being left with the responsibility is quite overwhelming unless I create dynamic encounters or make it all up on the spot. I'm often left having no idea what to do.  Examples of Why I Think I Need Help  This question isn't about these examples so please don't dig into them in the answers - they're here to explain the basic things that leave me feeling like I don't know how to proceed as a GM.  I've read preparation is important, but last time I tried to make a story, the players immediately killed the primary character (a friendly person.) within about 2 minutes of starting the game. While I realize now that I let the dice take control over the story, it was rather difficult to continue a story that relied on this friendly character, with the players making enemies with the character's faction. It is still discouraging when my only other attempt at a prepared story ended with the players wanting to go in a complete opposite direction of where I planned. Both times, the hours of work I put into it were completely wasted, leaving me as ignorant as I always was before.  My group often wants to play Shadowrun, Earthdawn, or L5R- but I feel as though I am a horrible GM. The last time we played Shadowrun, I was bored out of my mind as we rolled hundreds of dice to do an obvious straight-forward "kick down the door" mission.   If I make security as I believe it should or as the forum threads describe their security or a few example missions I can scrounge up (examples of stories, not actual gameplay), my players literally feel completely hopeless as to how to achieve their mission without total failure.   One time when I was a teenager, I had players blow up at me saying "We have no idea what would work. You're telling us that our character's experience or the ally operative says that would get us killed, but what else can we do?" Since there are no examples of what players can do - I am left with the same feeling as the players. As an adult, I've had a similar occurrence where a player said to me, "I don't know what to do. I don't think we can even begin this mission." I had to make something up and just go with it, creating an NPC group (that fit in with the story) that goes in before the players to do something stupid so they don't have to. Mainly because I ALSO didn't know how to accomplish the mission, so I acted as the players would using my NPC group. Extremely boring and IMO a failed dungeon run.  I've had successful stories, but I've had more boring encounters than successful ones. Seeing as how roleplaying takes endless hours to complete a story, I do not see benefit in an activity that wastes more time than is enjoyable.  How Do You Learn To GM?  Where are the resources? Books, blogs, gameplay examples? Am I missing them? How did YOU learn to be a good GM? Or do you feel as bad at this as I do, even when it's a successful, fun encounter? Does Time Machine backup programs like Photoshop and iWork? I'm looking to view and archive all my old DMs in Twitter, since I would like to revive conversations that took place 1 year ago.  Acording to Twitter they don't delete old messages, they simply archive them in their database.  Is there a way to view and download all of these messages, just like you can view all your old public tweets with third party apps?  UPDATE: I found  but apparently it's not working for DMs now Factorial identity $n!=1+(1-1/1!)n+(1-1/1!+1/2!)n(n-1)+\cdots$ Show that $\displaystyle{n!=1+\left(1-\frac1{1!}\right)n+\left(1-\frac1{1!}+\frac1{2!}\right)n(n-1)+\cdots}$. I can't figure out how this can be solved. I tried to use the binomial theorem but I couldn't prove it. Any help will be greatly appreciated. Below you find the design for our business cards. The triangles are yellow, as you can see outside of the canvas. However, inside the canvas, only when I set the Color Mode to CMYK, they appear pink. When I switch the Color Mode back to RGB, they appear yellow as expected. However, when I print them, in both cases they turn out pink...   I would just like to know what is going on here and how I can change them back to the right colors?  CMYK Colors: