I’ve found this problem in a number theory book, it says: If $a \equiv b \pmod n$ with $c&gt;0$ and $\gcd(c,n)=1$, prove that $$\frac{a}{c} \equiv \frac{b}{c} \pmod n$$ It seems trivial but unfortunately i don’t know how to prove it . I don’t know is it necessary to $c\mid a,b$ Because if $c\nmid a,b \implies \frac{a}{c} \notin \mathbb Z$ But modular arithmetic deals only with integers (I’m new to congruences  so maybe I’m wrong.)
I have seen other criteria for divisibility by 7. Criterion described below present in the book Handbook of Mathematics for IN Bronshtein (p. 323) is interesting, but could not prove it. Let $n = (a_ka_{k-1}\ldots a_2a_1a_0)_{10} = \displaystyle{\sum_{j=0}^{k}}a_{k-j}10^{k-j}$. The expression $$ Q_{3}^{\prime}(n) = (a_2a_1a_0)_{10} - (a_5a_4a_3)_{10} + (a_8a_7a_6)_{10} -\ldots  $$ are called alternating sum of the digits of third order of $n$. For example,  $$ Q_{3}^{\prime}(123456789) = 789-456+123=456 $$ Proposition: $7 | n \ \Leftrightarrow \ 7 | Q_{3}^{\prime}(n)$.  proof. ??  Thanks for any help.
While browsing on Safari and listening to iTunes I come across a video and then press the "play" button on the keyboard (to stop the music while I watch the video) it starts/stops the video instead of the music (I have no control over the music).  How can I quickly play/pause my music without affecting the video? Root of unity for $n+1$ Let $n \ge 1$ and $u_{1},u_{2},...,u_{n}$ are all root of unity for $n+1$ and there are different from $1$.(a) Find polynomial $f \in \mathbb R[x]$ of degree $n$ for which: $f(u_{k})=0$ for $k=1,2,...,n$.(b) Calculate real coefficient and imaginary part for $(2-u_{1})(2-u_{2})\cdot...\cdot(2-u_{n}).$    I have a knowledge about root of unity but I completely don't know how to do it because I think that I need some extra trick to do it but my ideas finished. I know that $\varepsilon_n^{(k)} = \cos\left(\tfrac{2k\pi}{n}\right) + i\sin\left(\tfrac{2k\pi}{n}\right) = e^\frac{2\pi i k}{n}$ and for $n&gt;1$ $\sum _{{k=0}}^{{n-1}}e^{{\frac  {2\pi ik}{n}}}=0$ but I think it is too little knowledge to solve this task.Can you help me? Does the derivative of a continuous function goes to zero if the function converges to it? In how many ways can a number be expressed as a sum of consecutive numbers? I'm wondering if there's any log which stores information about a particular program such as when the program has started and closed manually on Windows such as MS Word, or Chrome. Any ideas? Which equals operator (== vs ===) should be used in JavaScript comparisons? I'm using  to go through JavaScript, and it's returning many suggestions to replace == (two equals signs) with === (three equals signs) when doing things like comparing idSele_UNVEHtype.value.length == 0 inside of an if statement.  Is there a performance benefit to replacing == with ===?   Any performance improvement would be welcomed as many comparison operators exist.  If no type conversion takes place, would there be a performance gain over ==? How to avoid nested With[]? Pivot in SQL 2008 R2 Default poster size What is the typical size for a conference poster? Is it "ask the organizers", or can I safely assume that I'll get a default size, and that there will be enough space to hang it at the poster session? The description of the  says:     Your Intelligence score is 19 while you wear this headband. It has no   effect on you if your Intelligence is already 19 or higher.   The  spell description says that my Intelligence score becomes 1 on a failed save:     [...] On a failed save, the creature's Intelligence and Charisma   scores become 1. The creature can't cast spells, activate magic items,   understand language, or communicate in any intelligible way. [...]   If I am affected by Feeblemind and I am wearing the Headband of Intellect, would I have an Intelligence score of 19 or 1?   Also, what about the other effects like not being able to cast, activate magic items, understand languages or communicate in any way? Those seem Intelligence-related, but are mentioned additionally to the reduction.   If the Intelligence score becomes 19, are the other effects still active? System.CalloutException: You have uncommitted work pending I have same exception can please   The previous post issue is resolved.  Exception: :System.CalloutException: You have uncommitted work pending. Please commit or rollback before calling out  Visualforce Page   &lt;apex:page controller="Controller" action="{!init}" showHeader="false" sidebar="false"&gt;     &lt;center&gt;         &lt;apex:pageBlock title="Request Listener"&gt;&lt;/apex:pageBlock&gt;               &lt;/center&gt; &lt;/apex:page&gt;   Controller :            PUBLIC with sharing class Controller {        PUBLIC String fromNumber      = ApexPages.currentPage().getParameters().get('From');     PUBLIC String toNumber        = ApexPages.currentPage().getParameters().get('To');     PUBLIC String body            = ApexPages.currentPage().getParameters().get('Body');        PUBLIC PageReference init()     {                    System.debug('From Phone Number :' +fromNumber);         System.debug('To phone NUmber :' + toNumber);         System.debug('Message Body :' + body);          TwilioRestClient Client = TwilioAPI.getDefaultClient();                                                                SYSTEM.DEBUG('FROM AND To  Number is NOT NULL');                      String formattedNumber='+919876543210';                 IF(body != NULL)                 body = body;             ELSE                 body = '';             Case c = NEW Case (Subject = formattedNumber,Description = body,Origin = 'Phone');             INSERT c;                               Map&lt;String,String&gt; params1 = new Map&lt;String,String&gt;             {                'To'   =&gt; fromNumber,                'From' =&gt; '+1908280****',                'Body' =&gt; 'Conformation to customer'             };             TwilioMessage message = client.getAccount().getMessages().create(params1); /* Valid conformation SMS sent to the Customer.*/                                            return null ;     } }   While loin as admin then click preview on visualforce page executed but case is not created.I have checked the the debug log it throw the exception. Is $$d_1\mid n,d_2\mid n\iff [d_1,d_2]\mid n$$ true ? And if yes, how can I prove it ? I recall that $[d_1,d_2]$ is the least common multiple.  My tries  For the implication : Let $d_1\mid n$ and $d_2\mid n$. Then, $d_1d_2\mid n^2$. By the way, since $d_1,d_2\mid n$, we have that $(d_1,d_2)\mid n$ where $(d_1,d_2)=:\gcd(d_1,d_2)$  and thus $$[d_1,d_2](d_1,d_2)\mid n^2.$$  Question: How can I get $[d_1,d_2]\mid n$ from this?  For the converse, since $d_1,d_2\mid [d_1,d_2]\mid n$, the claim follow. Overheating results in system shutdown The code for reference is this  @article{matcontcl,  title= {MATCONT and CL MATCONT},  author = {A. Dhooge, W. Govaerts, Yu.A. Kuznetsov, W. Mestrom, A.M. Riet and  B.Sautois},  year = {2006},  journal = {Continuation toolboxes in matlab}, }   This is the error that follows  This is BibTeX, Version 0.99d (TeX Live 2015/W32TeX) The top-level auxiliary file: PropuestaInvestigacion.aux The style file: abbrv.bst Database file #1: biblio.bib Too many commas in name 1 of "A. Dhooge, W. Govaerts, Yu.A. Kuznetsov, W. Mestrom, A.M. Riet and B.Sautois" for entry matcontcl while executing---line 1049 of file abbrv.bst Too many commas in name 1 of "A. Dhooge, W. Govaerts, Yu.A. Kuznetsov, W. Mestrom, A.M. Riet and B.Sautois" for entry matcontcl while executing---line 1049 of file abbrv.bst Too many commas in name 1 of "A. Dhooge, W. Govaerts, Yu.A. Kuznetsov, W. Mestrom, A.M. Riet and B.Sautois" for entry matcontcl while executing---line 1090 of file abbrv.bst Too many commas in name 1 of "A. Dhooge, W. Govaerts, Yu.A. Kuznetsov, W. Mestrom, A.M. Riet and B.Sautois" for entry matcontcl while executing---line 1090 of file abbrv.bst (There were 4 error messages)   How could I fix it? From a toggle to an impulsion I'm trying to make a circuit (with no programmable circuits, only logic gates, flip flops and latches) that outputs a short impulsion each time it's input toggles from 0 to 1. Do you have any idea of how to do so? Looking for intro to Conformal Bootstrap We were given the following problem as an assignment, we proved the converse in class for prime numbers; however, we can't use the assumption that $\operatorname{gcd}(a,b) = 1$.  So I am struggling to think of my first possible move.  We are given $ar + bs = 1$, the only think I can think of is let $d$ be a common divisor or $ar + bs$ and then show that the $d = 1$.  However, I am not really sure how to do that.  If anyone could just give me a hint or two to nudge me in the right direction I would be appreciative. I need to match a space character in a PHP regular expression. Anyone got any ideas?  I mean like "gavin schulz", the space in between the two words. I am using a regular expression to make sure that I only allow letters, number and a space. But I'm not sure how to find the space. This is what I have right now:  $newtag = preg_replace("/[^a-zA-Z0-9s|]/", "", $tag); In C++, what is the purpose of the scope resolution operator when used without a scope? For instance:  ::foo(); Show that a continuous function has a fixed point Question: Let $a, b \in \mathbb{R}$ with $a &lt; b$ and let $f: [a,b] \rightarrow [a,b]$ continuous. Show: $f$ has a fixed point, that is, there is an $x \in [a,b]$ with $f(x)=x$.  I suppose this has to do with the basic definition of continuity. The definition I am using is that $f$ is continuous at $a$ if $\displaystyle \lim_{x \to a} f(x)$ exists and if $\displaystyle \lim_{x \to a} f(x) = f(a)$. I must not be understanding it, since I am not sure how to begin showing this... Should I be trying to show that $x$ is both greater than or equal to and less than or equal to $\displaystyle \lim_{x \to a} f(x)$ ?