Is there any relation between $a\equiv x\pmod{b}$ and $a\equiv x\pmod{c}$? I saw a question which asked the remainder of $18!$ when it is divided by $437.$ I tried to solve it using the Wilson's Theorem. Since $437=19\times23$ which both are prime, I can rewrite it to be $$18!\equiv-1\pmod{19}\text{,}\quad22!\equiv-1\pmod{23}.$$ Then, since $22!=22\cdot21\cdot20\cdot19\cdot18!,$ and $22!\equiv-1\pmod{23},$ $21!\equiv-2\pmod{23},$ $20\equiv-3\pmod{23},$ $19\equiv-4\pmod{23},$ I can rewrite it into $$(-1)\cdot(-2)\cdot(-3)\cdot(-4)\cdot18!\equiv-1\pmod{23}\iff24\cdot18!\equiv-1\pmod{23}.$$ Since $24\equiv1\pmod{23},$ then $18!\equiv-1\pmod{23}$. Now, I'm stuck here. I got that $18!\equiv-1\pmod{19}$ and $18!\equiv-1\pmod{23}$. Are there any relations between those two? Any helps would be very appreciated.
Prove $x \equiv a \pmod{p}$ and $x \equiv a \pmod{q}$ then $x \equiv a\pmod{pq}$ Prove $x \equiv a \pmod{p}$ and $x \equiv a \pmod{q}$ then $x \equiv a\pmod{pq}$ for $p\neq q$ distinct primes.  Where can I start with this proof? It looks similar to the Chinese Remainder Theorem, but that deals with two different a values.
Sugar as a defence against bacteria? mentions that sugar "at a certain level acts as a preservative". I've always been taught that microorganisms eat sugar and expel acids, that is why sugary food are damaging to teeth. How is it that sugar acts as a preservative, then?  Googling the question I've found   though I don't credit the sources as being especially reputable. What is the real reason that sugar can act as a preservative? Ampscript date range lookup I want to use LookupOrderedRows and return rows between a start and end date (last month range). What is the best way to handle using Ampscript?   I can add one date to the where using   "DateField", @varStartDate,   But if I add the below to add additional where filters  "DateField", @varStartDate,"DateField", @varEndDate,   I get this error:     "Duplicate Key Field Name: DateField"   EDIT: Also I tried the range as the var for the LookupOrderedRows using   set @strRange = StringToDate(Format(DateDiff(@strStartDate, @strEndDate, "M"),"MM/dd/yyyy"))  and then in the LookupOrderedRows just using  "DateField", @strRange  But that got an error as well. Resuming a list Unable to send e-mail. Contact the site administrator if the problem persists When my web form is filled and submitted, I get redirected to a page that says:     Thank you for submitting.   but theres a red warning box there that says:     Unable to send e-mail. Contact the site administrator if the problem persists.   I think its the SMTP server, but I don't know how to fix that. If  $\mathbb R^*$ and $\mathbb C^*$ be respectively multiplicative groups of non-zero real and complex number then whether $\operatorname{Hom}(\mathbb Z_6,R^*\oplus C^*)$  is  isomorphic to $\mathbb Z_{12}$? or $\mathbb Z_6$? or $\mathbb Z_6\oplus \mathbb Z_2$? Failed to upload an image, "An error occurred at imgur" Currently I'm unable to upload an image to anywhere on site: e.g. posts or chat.                Is this problem with Imgur? What's the use/meaning of the @ character in variable names in C#? I discovered that you can start your variable name with a '@' character in C#.  In my C# project I was using a web service (I added a web reference to my project) that was written in Java. One of the interface objects defined in the WSDL had a member variable with the name "params". Obviously this is a reserved word in C# so you can't have a class with a member variable with the name "params". The proxy object that was generated contained a property that looked like this:  public ArrayList @params {     get { return this.paramsField; }     set { this.paramsField = value; } }   I searched through the VS 2008 c# documentation but couldn't find anything about it. Also searching Google didn't give me any useful answers. So what is the exact meaning or use of the '@' character in a variable/property name? How do I make my screenshot ask me what to do next every time? I take screenshots a lot. I usually take screenshot of an area, I then recheck if I have the desired area and then copy it to clipboard.  I want to change this behaviour so that immediately upon  pressing Shift+PrntScr the screen capture application allows me to select an area, and immediately afterwards it provides me with the option to save to a particular location or clipboard.  I want Shift+PrntScr to bring up a pop-up screen like this: How do I declare and initialize an array in Java? finding rational roots of polynomials Can I manually put an external hard disk to sleep? I'm sure there must be a simple reason for this but I can't seem to be able to import a shp file or a feature class from another file or sde geodatabase into a Feature Dataset without getting an error.  The feature classes come in fine if I don't don't try to import them into a Feature Dataset.  The feature dataset definitely has the same projection as the feature classes I'm trying to import so I don't think that's the problem.  The Error I get is ERROR 000210 and just says Failed to convert and cannot create output and Failed to execute (CopyFeatures).    Any ideas what I may be doing wrong here.  Seems like something so simple.  I'm trying to use the Feature Datasets to better organize my data which I think is there purpose.   Update:  So as you can see below, Jason helped me find the answer.   As suggested by Matt below, I added a new Idea to the ArcGIS Ideas site.  Here is the link to that, so please promote this up if you agree a better error message is needed here: I was hoping I would find some people in the industry that could help me decide what's the best price to charge to design (in this case) business cards.  What are standard prices for professional designers?  From a firm point of view? From a freelancer?  I don't want to overcharge clients but I don't want to ruin the industry by underselling my services. Number of elements in a finite $\sigma$-algebra How can I prove the following?  $\gamma (t)$ is unit speed, $\dot \gamma(t) \not= 0 \Rightarrow \ddot \gamma(t)$ is perpendicular to $\gamma(t)$   I don't really see where a problem would arise when $\dot \gamma(t)=0$ would cause such a problem Can every even integer be expressed as the difference of two primes? If so, is there any elementary proof? When is $(p - 1)! + 1$ a power of $p$? A friend asked me this question:     If $p$ is a prime, prove that $(p - 1)! + 1$ is a power of $p$ if and only if $p = 2, 3$ or $5$.   Clearly one direction is obvious, namely that $p=2,3,5$ implies $(p - 1)! + 1$ is a power of $p$.  The other direction is not clear to me. Since by Wilson's theorem $p$ divides $(p - 1)! + 1$ so we need to show that if there are no other prime factors then $p=2,3,5$. Can someone give me a hint for establishing this?  Thanks Isn't the word "experience" wrongly used in this context? Friend: Have you ever flirted with a female cop?      Me: Nope.      Friend: I have done that experience.   I think experience is the wrong word here. What else should he have said? The functions of each pair are inverse to each other. For each pair, check that both compositions give the identity function.  $F: \mathbb{R} \to \mathbb{R}$ and $F^{−1}:\mathbb{R} \to \mathbb{R}$ are defined by $F(x)=3x+2$ and $F^{−1}(y)=\dfrac{y−2}{3}$. for all y ∈ R  My attempt:     Inverse Function  For each particular but arbitrarily chosen $y \in \mathbb{R}$, according to the definition of $f^{-1}$, $f^{-1}(y) = \dfrac{y-2}{3}$ is a unique real number $x$ such that $f(x) = y$. \begin{align*} F(x) &amp; = y\\ 3x + 2 &amp; = y\\ x &amp; = \frac{y-2}{3} \end{align*} Therefore, $f^{-1}(y) = \frac{y-2}{3}$.  Compositions of Functions  The functions $g \circ f$ and $f \circ g$ are defined as follows: $$(g \circ f)(x) = g(f(x)) = g(3x + 2) = 3x + 2$$  for all $x \in \mathbb{Z}$. "Duplicate of...[deleted post]"