With,  $$U(n) = \lbrace k : (k, n) = 1 \space and \space 0 &lt; k &lt; n \rbrace.$$  For some $n$, each element of $U(n)$ will have itself as its own multiplicative inverse. As an example, for $n = 8$:  $$U(8) = \lbrace 1, 3, 5, 7 \rbrace$$ Inverse of $1, 3, 5, 7$ under multiplication modulo $8$ is respectively $1, 3, 5, 7$. And it is very weird, because in this case multiplication of $a$ with $b$ is same as division of $a$ with $b$.  My question is are there infinitely many $n$ such that $U(n)$ satisfies the property above? I wrote a script and tested this for all $n \lt 1000$ and found only the following solutions:  $$n = 1, 2, 3, 4, 6, 8, 12, 24$$  Are there any more solutions, or are they the only ones? How to show that?
I have a hard time formulating proofs.  For this problem, I  can see that if $n$ is equal to $8,$ this statement is true.  $(\mathbb{Z}/8\mathbb{Z})^{\times}$ includes elements:  $1,3,5,7$, and all of these are roots of $1-x^2 \pmod 8.$  And obviously $8$ divides $24.$  But how do I prove this without depending on number calculations and only using theorems?  Help Please?  I need a step by step walk through of how to do this proof and what theorems would be appropriate to use.
If I multiclass into 2 or more classes/subclasses that can cast spells, how do I determine what spells I know or can prepare?  For instance, if I am playing a multiclassed character with 1 level in warlock, 2 levels in bard, and 3 levels in paladin, how do I determine which spells I know/have prepared?    Related: Reflection is quad dependent and looks non realistic Im rendering varous objects with a Sharp glossy effect but the reflections are quad dependent on the object (image below), meaning they behave separately within every quad making it look 8bit-like. Can anyone tell me why this happens?. (Im pretty sure subdividing would fix it but I want to know why I cant have a low poly object reflecting normaly.  The object has 2 modifiers: Subdiv Surface and Solidify. (Perhaps the Subdiv is causing it?). Help me you Blender savys I dont have a powerful CPU to test and test within every render would take me hours and hours.  Image: I have a geotiff raster image that has a coordinate system with longitudes from 0 to 360. The horizontal center of the image is 180 longitude. See image below:    I want to transform it to EPSG:4326 SRS with -180 180 longtitude range.  And I want the center of the image to be at Greenwich meridian (0). I guess this srs is very widely used. I expect the result look like this:    So I use a gdalwarp command to reproject:  gdalwarp -s_srs '+proj=latlong +datum=WGS84 +pm=180dW' -t_srs EPSG:4326 test_col.tif test_4326.tif   But I only get a tiff with bigger dimensions (more pixels) and EPSG:4326 metadata. The image itself looks the same, as the initial one. But I expect it to swap the hemispheres.  How do I gdalwarp an image to be strictly -180 180 EPSG:4326 with the center in 0 longitude?  This is gdalinfo of my initial file:  Origin = (-0.102272598067084,89.946211604095552) Pixel Size = (0.204545196134167,-0.204423208191126) Metadata:   AREA_OR_POINT=Area Image Structure Metadata:   INTERLEAVE=BAND Corner Coordinates: Upper Left  (  -0.1022726,  89.9462116) (  0d 6' 8.18"W, 89d56'46.36"N) Lower Left  (  -0.1022726, -89.9462116) (  0d 6' 8.18"W, 89d56'46.36"S) Upper Right (     359.897,      89.946) (359d53'50.18"E, 89d56'46.36"N) Lower Right (     359.897,     -89.946) (359d53'50.18"E, 89d56'46.36"S) Center      ( 179.8975000,  -0.0000000) (179d53'51.00"E,  0d 0' 0.00"S)   This is gdalinfo after gdalwarp     Origin = (-180.102727401932952,89.946211604095552) Pixel Size = (0.091397622896436,-0.091420837939082) Metadata:   AREA_OR_POINT=Area Image Structure Metadata:   INTERLEAVE=PIXEL Corner Coordinates: Upper Left  (-180.1027274,  89.9462116) (180d 6' 9.82"W, 89d56'46.36"N) Lower Left  (-180.1027274, -89.9699975) (180d 6' 9.82"W, 89d58'11.99"S) Upper Right ( 179.8211116,  89.9462116) (179d49'16.00"E, 89d56'46.36"N) Lower Right ( 179.8211116, -89.9699975) (179d49'16.00"E, 89d58'11.99"S) Center      (  -0.1408079,  -0.0118929) (  0d 8'26.91"W,  0d 0'42.81"S) I am trying to hide Authoring information, Revision information, Publishing options tabs in create/edit node for any content type. I tried different modules with no luck. Please give me suggestions how to achieve this. Thanks a lot Why do ICs have NC (No Connection or No Function) pins? Including section titles in fancyhf headers Create a shortcut for URL? How can I create a shortcut for URL in my desktop? I want to use them in Firefox or Chrome.   If a copy and paste didn't work.   What script I have to use? How do I clear Bash's cache of paths to executables? Let $x &gt; 1$ and $a$, $b$ be positive integers. I know that $a$ divides $b$ implies that $x^a - 1$ divides $x^b - 1$. If $b = qa$, then  $$x^b - 1 = (x^a)^q - 1^q = (x^a - 1)((x^a)^{q-1} + \ldots + x^a + 1).$$  I'm interested in the converse of the statement. If $x^a - 1$ divides $x^b - 1$, does this imply that $a$ divides $b$? $f(f(x)f(y))+f(x+y)=f(xy)$ Find all functions $f\colon \mathbb R\rightarrow \mathbb R$ such that for all reals $x$ and $y$:   $$f(f(x)f(y))+f(x+y)=f(xy).$$   It was six hours ago in IMO 2017 (problem 2).  I tried  the standard way: $x=0$, $x=y$, $x=1$,etc. but without any success.   Let $f(0)=a$. Hence, for all $x\in\mathbb R$ we have $f(a(f(x))+f(x)=a$. What is the rest?   Also we have $(x-1)(y-1)-1+x+y-1=xy-1$ and  $1-(1-x)(1-y)+1-x-y=1-xy$. Error after installing search service application I got this this error after setting up Search Service Application with Powershell.     The search service is not able to connect to the machine that hosts the   administration component. Verify that the administration component      'a23fd668-f823- 4ab2-b0fc-b0baebfd4af2' in search application   'SharePoint_Search_Service' is in a good state and try again.   The Sharepoint Search Service was showing error when I visited   Central Administration->Application Management->Service Applications->     Manage Service Applications  Here is the powershell script used  $App1 = "APP-Server-01" $APP2 = "APP-Server-02" $SearchAppPoolName = "SharePoint_SearchApp" $SearchAppPoolAccountName = "EXSERV\Administrator" $SearchServiceName = "SharePoint_Search_Service" $SearchServiceProxyName = "SharePoint_Search_Proxy" $DatabaseName = "SharePoint_Search_AdminDB"   Create a Search Service Application Pool  $spAppPool = New-SPServiceApplicationPool -Name $SearchAppPoolName -Account     $SearchAppPoolAccountName -Verbose   Start Search Service Instance on all Application Servers  Start-SPEnterpriseSearchServiceInstance $App1 -ErrorAction SilentlyContinue Start-SPEnterpriseSearchServiceInstance $App2 -ErrorAction SilentlyContinue Start-SPEnterpriseSearchQueryAndSiteSettingsServiceInstance $App1 -ErrorAction         SilentlyContinue Start-SPEnterpriseSearchQueryAndSiteSettingsServiceInstance $App2 -ErrorAction        SilentlyContinue   Create Search Service Application  $ServiceApplication = New-SPEnterpriseSearchServiceApplication -Partitioned -Name        $SearchServiceName -ApplicationPool $spAppPool.Name -DatabaseName $DatabaseName   Create Search Service Proxy  New-SPEnterpriseSearchServiceApplicationProxy -Partitioned -Name       $SearchServiceProxyName -SearchApplication $ServiceApplication How does the simulation hypothesis relate to the ethics of simulated violence? Say, for the sake of this question,  is real. Are we then hurting "persons"/conscious beings in "our simulations" - video games - when we fight them? Why (not)? If so, is this unethical? Think for example about shooting an alien in the game No Man's Sky.  Edit/answer on the comment of Joseph Weissman: My reasoning is like this:   It is unethical to harm a person (with a person being everything that has ). By the simulation hypothesis (assuming it's true), everything around us (everything that we know of), is simulated. By 1 and 2, all persons we know of are simulated. So it's unethical to harm the simulated persons around us. A game is a simulation just like we are in a simulation, so the game has simulated persons. Therefore -  - it's unethical to harm the persons in a video game.   I know that the premises - besides the simulation hypothesis itself - are debatable, but that's the reason I asked the question in the first place.  I was also thinking that the characters in the video game differ because they have far less to no intelligence, that much of it is scripted. But I neither have a proof that we ourselves aren't scripted, with free will being an illusion. Also, how intelligent does a character in a simulation/video game has to be to have personhood then? I think this is a case of the .  Just to be sure, please note that this question is not about whether the simulation hypothesis is true or how likely it is to be true. It is about (one of the) consequences if the simulation hypothesis is true. It's also not a question about violence in games in general. How to parse a string into a nullable int I am using my notes app on OSX 10.11.5 and iOS 8. For the most part, things are working, but I am noticing that many of my older notes are not searchable. When I type something in the search box, these older notes do not appear in search results even if there is a match. However, if I edit a note or create a new one, it is fully searchable.  My guess is that somehow the search indexes for my notes got corrupted. Is there any way to fix this or rebuild the indexes? I am trying to implement the following K-map with a table    AB C     00 01 11 10       ______________     0| 1  0  1  0     1| 0  1  1  0   I implemented it like this:   \begin{table}[htdp] \begin{center} \begin{tabular}{c|c|c|c|c|}  &amp;00 &amp; 01&amp; 11&amp; 10 \\ \hline 0&amp;1&amp;1&amp;1&amp;1\\ 1&amp;0&amp;0&amp;0&amp;0\\ \end{tabular} \end{center} \label{default} \end{table}   However on the top row I have vertical lines that I'd like to not see. I went with this approach because if i was to use a one row with the information so that I wouldn't have to worry about aligning the headings. So is there a way I can get the vertical lines not to appear in the heading row? How can I implement "cut outs" for lighting in OpenGL? So, I'm working with OpenGL (I'm not exactly sure of the version), and I want to do an old-style lighting setup by essentially drawing a black rectangle over the screen, and drawing white circles over the areas that should be lit. I already have the rectangle and the circles, but I can't figure out how to make it so that the white circles are transparent when applied to the black areas. Any ideas? I tried to use blending modes, but it still evades me...  EDIT: Here's the code that I'm using (Python).  view_buf = Buffer(GL_INT, 4) glGetIntegerv(GL_VIEWPORT, view_buf) view = view_buf.to_list() if hasattr(view_buf, "to_list") else view_buf.list  glDisable(GL_DEPTH_TEST)  glDisable(GL_LIGHTING)  glEnable(GL_BLEND)  if logic.lights['smooth']:     glEnable(GL_LINE_SMOOTH)  # # Make sure we're using smooth shading instead of flat # glShadeModel(GL_SMOOTH)  # Setup the matrices  glMatrixMode(GL_PROJECTION) glPushMatrix() glLoadIdentity() gluOrtho2D(0, view[2], 0, view[3]) glMatrixMode(GL_MODELVIEW) glPushMatrix() glLoadIdentity()  #glAlphaFunc(GL_GREATER, 0.5)  glColor4f(0.0, 0, 0, 0.75)  glBegin(GL_QUADS);                              # Draw Black screen glVertex2d(0.0, 0.0); glVertex2d(view[2], 0.0); glVertex2d(view[2], view[3]); glVertex2d(0.0, view[3]); glEnd()  glColor4f(1, 1, 1, 1)  degtorad = 3.14159/180.0   glBlendFunc(GL_ZERO, GL_SRC_ALPHA)  def DrawCircle(x, y, radius, fill = 1, step = 11):      glVertex2f(x, y)      if fill:         glBegin(GL_TRIANGLE_FAN)     else:         glBegin(GL_LINE_LOOP)      for a in range(0, 360, step):         angle = degtorad * a         glVertex2f(x + (sin(angle) * radius), y + (cos(angle) * radius))      glEnd()   #for layer in range(len(logic.lights['layers'])): #   size =  for light in logic.lights['points']:     DrawCircle(light[0][0] * view[2], (1 - light[0][1]) * view[3], light[1], step = light[2])     if logic.lights['smooth']:         DrawCircle(light[0][0] * view[2], (1 - light[0][1]) * view[3], light[1], 0, light[2])  # Reset the state glPopMatrix() glMatrixMode(GL_PROJECTION) glPopMatrix() I often read that one should not place swap partitions on a SSD drive, as this may harm the device. Is this true? Can you please explain the reason to me?  Because I otherwise would have thought that placing swap on an SSD is the best choice, as it's much faster than HDDs and therefore swapping RAM contents to the SSD is not as slow as it would be with the HDD... Entire function and the series of its successive derivatives I recently came across the result that if $g$ is holomorphic in a neighbourhood of a point $z_0$ and  $$\sum_{n=0}^{\infty }{g^{(n)}(z_0)} $$ converges then $g $ is an entire function. Now, my doubt is if the above series is absolutely convergent then can we say something more about $g$, like say $g$ is a polynomial? I could not find any non-polynomial  entire function( I tried with functions like $e^z$, $\sin z $, et al), whose successive derivatives converge absolutely as series. Any help please? Kernel requires an x86-64 CPU, but only detected an i686 CPU. How can I install Lubuntu/Ubuntu? I'm trying to install Lubuntu on my laptop (sorta old 2005). [Laptop stats: Intel® Celeron® CPU 410 @ 1.46 GHz 1.47GHz, 448 MB of RAM]  I press Install and this message appears:   This kernel requires an x86-64 CPU, but only detected an i686 CPU.  Unable to boot - please use a kernel appropriate for your CPU.   What can I do to install Lubuntu on my computer? Second law of Newton for variable mass systems Frequently I see the expression  $$F = \frac{dp}{dt} = \frac{d}{dt}(mv) = \frac{dm}{dt}v + ma,$$ which can be applied to variable mass systems.  But I'm wondering if this derivation is correct, because the second law of Newton is formulated for point masses.   Furthermore if i change the inertial frame of reference, only $v$ on the right side of the formula $F = \frac{dm}{dt}v+ma$ will change, meaning that $F$ would be dependent of the frame of reference, which (according to me) can't be true.   I realize there exists a formula for varying mass systems, that looks quite familiar to this one, but isn't exactly the same, because the $v$ on the right side is there the relative velocity of mass expulsed/accreted. The derivation of that formula is also rather different from this one.   So my question is: is this formula, that I frequently encounter in syllabi and books, correct? Where lies my mistake.