Suppose k is an integer such that $6k + 1$, $12k + 1$, $18k + 1$ are primes. Show that $n = (6k + 1)(12k + 1)(18k + 1)$ is a Carmichael number.  This is a question on a past exam that I find difficulty to answer. Can anyone help?
Let $t∈\Bbb N$ and let $x=6t+1, \:y=12t+1$ and $z=18t+1$. $x, y$ and $z$ are all primes and let $n=xyz$. Prove that  $a ^{n-1} \equiv 1 \pmod  n\;$ whenever $ a∈\Bbb Z$ and $\gcd(a,n) = 1$.  I have started using the Euler-Fermat Theorem since $\gcd(a,n)=1$. Giving $\phi (n)=(1296t^3) $ But unsure how to proceed?
What should I consider when choosing continuous lighting vs. strobes for studio photography? Why has Steve Nash moved to the LA Lakers in sign-and-trade deal? Congratulations to Steve Nash on his move to Lakers !  His contract with the Suns has expired so I don't understand why he moved in a sign-and-trade deal.    Why resign and move in trade? Why didn't the Lakers sign Nash without any trade? What the reason the number of lines differs?  $ head -n 100000 ./access.log &gt; ./data/log.sample $ cat $_ | wc -l 1933424 are the uses of 'bare infinitive' & 'participle' same here? I saw him sing     I saw him singing    In the first example, the verb sing is used as a bare infinitive and in the following one it is exemplified as a present participle. My question is  if both of them are expressing the same meaning? My wife has several files and folders that somehow ended up having filenames that have caused them to be undeleteable (can't be deleted) by normal means or via the command line. I believe the filenames are too long due to the depth of the folder structures. Does anyone know of a good utility for cleaning up files like this? I have a bunch of raw data values that are dollar amounts and I want to find a confidence interval for a percentile of that data.  Is there a formula for such a confidence interval? biblatex - issue number of journal in date I am building a bibliography using biblatex and biber (\usepackage[backend=biber,minnames=2,backref,firstinits,sortcites]{biblatex}) with a lot of cited papers. Therefore I often have the case of a journal name, volume, issue, month and year, all necessary.   Now the strange thing is, if I have a bibitem such as:  @article{item,   title = {Some Nice Scientific Title},   author = {Doe, Jane},   journal = {Important Journal},   volume = {50},   issue = {20},   pages = {1877--1879},   year = {1984},   month = {Jun},   publisher = {Important Society},   doi = {00.0000} }   the bibliography looks like this:    Can you spot it? The issue number 20 is in the brackets with the date. Now it looks like it is the day in the date.  So, how do I get around this?  Here's the full MWE:  \documentclass{scrartcl}  \usepackage[backend=biber,minnames=2,backref,firstinits,sortcites]{biblatex}  \usepackage{filecontents} \begin{filecontents}{item.bib}   @article{item,     title = {Some Nice Scientific Title},     author = {Doe, Jane},     journal = {Important Journal},     volume = {50},     issue = {20},     pages = {1877--1879},     year = {1984},     month = {Jun},     publisher = {Important Society},     doi = {00.0000}   } \end{filecontents}  \addbibresource{item.bib}  \begin{document}  \printbibliography[title = {References},heading=bibintoc] \nocite{*}  \end{document} What is the most efficient way to write the old-school:  StringBuilder sb = new StringBuilder(); if (strings.Count &gt; 0) {     foreach (string s in strings)     {         sb.Append(s + ", ");     }     sb.Remove(sb.Length - 2, 2); } return sb.ToString();   ...in LINQ? How is the no-cloning theorem compatible with the fact that fan-out gates work? I have some difficulty with understanding no-cloning theorem. Simply speaking, according to the theorem, it is not possible to copy a quantum state.  On the other hand, CNOT gate can be used as so-called fan-out gate which purpose is to copy one qubit to another one, previously in state $|0\rangle$.  It seems that these two facts negate each other.   My question: How is no-cloning theorem compatible with the fact that fan-out gate works? File access time not updating in ubuntu 12.04 I would like to use Flash Player to show web contents but since today (12 Jan 2021) Adobe is blocking it. Is there a way to use it? I've tried Firefox, Chrome, Safari and Brave browsers. Forbid links to open new tabs in Google Chrome let $\gamma$ be a closed continuosly differentiable path in the upper half plane  not passing through $i$. What are the possible values of the integral $$\frac{1}{2\pi i}\int_{\gamma}\frac{2i}{z^2+1}dz$$ well the integral can be broken like  $$\frac{1}{2\pi i}\int_{\gamma}\frac{2i}{(z+i)(z-i)}dz=$$  $$\frac{1}{2\pi i}\int_{\gamma}\frac{dz}{z-i}dz-\frac{1}{2\pi i}\int_{\gamma}\frac{dz}{z+i}dz=$$ by Cauchy Integral Formuale $$f(i)-f(-i)$$,so the second integral is $0$ as it is analytic in the upper half plane, but the first integral iss just $n(\gamma,i)$, which iss the windding number of $\gamma$ around $i$ what more I can say? Thank you. Why does the collada exporter not export texture references? I've been having trouble exporting collada objects from blender, so I stripped everything back and textured a single cube, and exported it.  This is a completely trivial model, one cube, with one material assigned to it, the material has a single 'g.png' texture assigned to it.   The exported collada data looks like this:    &lt;library_materials&gt;     &lt;material id="Material-material" name="Material"&gt;       &lt;instance_effect url="#Material-effect"/&gt;     &lt;/material&gt;   &lt;/library_materials&gt;   Where:    &lt;library_effects&gt;     &lt;effect id="Material-effect"&gt;       &lt;profile_COMMON&gt;         &lt;technique sid="common"&gt;           &lt;phong&gt;             &lt;emission&gt;               &lt;color sid="emission"&gt;0.2879998 0.2879998 0.2879998 1&lt;/color&gt;             &lt;/emission&gt;             &lt;ambient&gt;               &lt;color sid="ambient"&gt;0 0 0 1&lt;/color&gt;             &lt;/ambient&gt;             &lt;diffuse&gt;               &lt;color sid="diffuse"&gt;0.2163267 0.1845119 0.2121569 1&lt;/color&gt;             &lt;/diffuse&gt;             &lt;specular&gt;               &lt;color sid="specular"&gt;0.5 0.5 0.5 1&lt;/color&gt;             &lt;/specular&gt;             &lt;shininess&gt;               &lt;float sid="shininess"&gt;50&lt;/float&gt;             &lt;/shininess&gt;             &lt;index_of_refraction&gt;               &lt;float sid="index_of_refraction"&gt;1&lt;/float&gt;             &lt;/index_of_refraction&gt;           &lt;/phong&gt;         &lt;/technique&gt;         &lt;extra&gt;           &lt;technique profile="GOOGLEEARTH"&gt;             &lt;double_sided&gt;1&lt;/double_sided&gt;           &lt;/technique&gt;         &lt;/extra&gt;       &lt;/profile_COMMON&gt;       &lt;extra&gt;&lt;technique profile="MAX3D"&gt;&lt;double_sided&gt;1&lt;/double_sided&gt;&lt;/technique&gt;&lt;/extra&gt;     &lt;/effect&gt;   &lt;/library_effects&gt;   This seems to be missing the texture data associated with the object.   By comparsion, exporting a wavefront object generates this .mtl file:  # Blender MTL File: 'basic.blend' # Material Count: 1  newmtl Material Ns 96.078431 Ka 0.000000 0.000000 0.000000 Kd 0.790000 0.790000 0.790000 Ks 0.500000 0.500000 0.500000 Ni 1.000000 d 1.000000 illum 0 map_Kd g.png   &lt;----- ie. Blender is aware of this texture   So... is the collada exporter for blender just broken, or does it require some kind of special config in order to function correctly?  If the latter, how/what is required to make this work?  (NB for what it's worth, the UV coordinates are exported in the .dae, abit they obviously are completely useless without a texture assigned to them; I've also tried adding two different textures to see if the exporter needs that because it naively assumes 'one texture = don't bother, it's one texture for the whole model'. That didn't work either) Consider the symmetric group $S_5$. I would like to find how many elements of $S_5$ are of order 5, and how many are of order 6. I would also like to determine what the maximum order of an element in this group would be.  Here is what I have so far: elements of order 5 are the 5-cycles. Elements of order 6 (since a 6-cycle is impossible for 5 elements) must have at least an even cycle and a cycle of length divisible by 3, and 2 + 3 = 5, so elements of order 5 are a combination of 2-cycles and 3-cycles.  How can I find the total count of such elements of order 5 and order 6, and the maximum order? I'm not entirely sure where to go from here. Double Factorial: Number of possibilities to partition a set of $2n$ items into $n$ pairs I know that the partition of $2n$ items into $n$ pairs has something to do with double factorial, but I am not sure how many possibilities we exactly have.   We can choose such a partition into pairs by first choosing $n$ elements to be in one half and the remaining $n$ in the other half, which gives ${{2n} \choose n}$ possibilities. Then we can take a permutation of both sets and match them accordingly.   This gives $${{2n} \choose n} n! n!=\frac{(2n)! n! n!}{n! n!}=(2n)!$$ possibilities. But with this we overcount, since we do not want the pairs to be ordered. Each of the $n$ pairs is counted twice, wherefore we have to divide by 2 for every pair, hence $\frac{(2n)!}{2^n}$?   Well, another possiblity to do that is just to iteratively pick $2$ elements among the remaining ones, which gives ${{2n} \choose 2} {{2n-2} \choose 2} \dotsb {4 \choose 2}$  I am not sure, but I do not think that this is the same and also, I do not see the connection to double factorial. Would be very glad if you could help me with these 3 possiblities.   Thank you. Is there a difference between "write" and "write down"? Need to get logfile smaller How can I get rid of those little white dots? I am using Blender 2.70a. My refrigerator line is 20 amp with 12 gauge wire. Can I tap into that with 14/2 wire, and run a light switch and 1 LED light?