I would like to prove it using Euler's Theorem, which states that if $\gcd(a,n)=1$,then $a^{\phi(n)}=1 \pmod{n}$. Then, $480$ should be factored, but I am unsure how to proceed with the proof. Thanks in advance.
I am trying to show that $a^{12} \equiv 1 \mod 32760$ for each $a \in \mathbb{Z}$ coprime to $32760$. I have found the prime decomposition $32760 = 2^3 \cdot 3^3 \cdot 5 \cdot 7 \cdot 13$ and determined $\phi(32760)$, where $\phi$ is the Euler's totient function. This gives me that $\phi (32760) = 8640$.   Using Euler's congruence, this gives that $a^{8640} \equiv 1 \mod 32760$, but this is still far from the result I want to show... I also know a consequence of Eulers congruence, which states that $e \equiv e' \mod \phi (n)$ implies that $a^e \equiv a^{e'} \mod n$, but I do not see how to use this...  Any hints would be appreciated.  $\textbf{EDIT:}$ I made a mistake in my primefactorization, it is edited now.
I want to show that if a function $f:[a,b]\rightarrow \mathbb R$ satisfies a Hölder condition of order $\alpha &gt; 1 $ then it is constant.  The way I think of it is as follows:  $$|f(x) - f(y)| &lt; K|x-y|^\alpha$$  $$\frac{|f(x) - f(y)|} {|x-y]}  &lt; K|x-y|^{\alpha -1}$$  $$\lim_{y\rightarrow x} \frac{|f(x) - f(y)|} {|x-y]} \le \lim_{y\rightarrow x}  K|x-y|^{\alpha -1} =0 $$ As the limit is $0$, we can remove the modulus, so we get:  $$\lim_{y\rightarrow x} \frac{f(x) - f(y)} {x-y} = 0$$ So $f$ is derivable and $f'(x) = 0$ for all $x$ in $[a,b]$. Note that the reason we can add the limit $y\rightarrow x$ is because $[a,b]$ is closed in $\mathbb R$.  However, the question gives as a hint using the mean value theorem. I am not sure why one should do that. You would first have to prove that $f$ is derivable in a similar manner to what I did, and then prove that $f$ is constant. Or is there a simpler way to do it and I am missing it?  Also please inform me of any mistakes I did in the proof (if any)/ Thank you! Help me understand adjusted odds ratio in logistic regression I've been having a hard time trying to understand the use of logistic regression in a paper. The paper available  uses logistic regression to predict probability of complications during cataract surgery.   What is confusing me is that the paper presents a model that assigns odds ratio of 1 to baseline described as follows:     A patient whose risk profile was in the reference group for all risk indicators (ie adjusted OR=1.00 for all in Table 1) may be regarded as having a ‘baseline risk profile’, and the logistic regression model indicates a ‘baseline predicted probability’ for PCR or VL or both=0.736%.   So the probability of 0.00736 is presented with odds ratio of 1. Based on the transformation from probabilities to odds ratios: $o=\frac{p}{1-p}$, this can't be equal to 1:  $0.00741=\frac{0.00736}{1-0.00736}$.  It gets even more confusing. The composite odds ratios that represents multiple covariates having values different than the baseline is used to calculate predicted risk.      ...the composite OR from Table 1 would be 1.28 X 1.58 X 2.99 X 2.46 X   1.45 X 1.60 = 34.5, and from the graph in Figure 1, we see that this OR corresponds with a predicted probability of PCR or VL or both of around 20%   The only way to arrive at the values the paper is giving as examples is to multiply the baseline probability with composite odds like this: $0.2025=\frac{(34.50\ \times\ 0.00736)}{1\ +\ (34.50\ \times\ 0.00736)}$.  So what is going on here? What is the logic for assigning odds ratio 1 to a baseline probability that is not 0.5? The update formula I've came up with above comes up with the right probabilities for examples in the paper but this is not the direct multiplication of odds ratio I'd expect. What is it then? Usage of italics in writing Electrostatics and tv screens Use chown to set the ownership of all a folder's subfolders and files? The characteristic and minimal polynomial of a companion matrix of a monic polynomial $f \in \mathbb F\left[x\right]$ in $1$ variable $x$ over a field $\mathbb F$ plays an important role in understanding the structure of finite dimensional $\mathbb F[x]$-modules.  It is an important fact that the characteristic polynomial and the minimal polynomial of $C(f)$ are both equal to $f$. This can be seen quite easily by induction on the degree of $f$.  Does anyone know a different proof of this fact? I would love to see a graph theoretic proof or a non inductive algebraic proof, but I would be happy with anything that makes it seem like more than a coincidence! I have two given names and one surname. Is it a problem if I omit the surname from my ticket? I have two names as my given name in the passport and I typically just use this as my first and last name. However, my passport has a surname too which I almost never use. Due to this, I mistakenly provided my given name as first and last name.   As per the passport,  Surname : AAAA Given name(s) : BBBB CCCC   In the ticket, First name : BBBB Last name : CCCC  The credit card I've used to make this booking has my name as BBBB CCCC.  Will this create a problem at the travel time?  Update : the airline says that they can't update the name in the ticket. They are saying that since they have made a remark of the passport details boarding will not be a problem but I might face problems at immigration. Do they check name details in itenerary to exactly match passport at the time of immigration? Is that true in your experience?  Update : Travelled without any problem, neither at boarding gate, immigration nor after arrival at border police control. I have seen other similar questions but they don't seem to be what I want, please correct me if I'm wrong.  For simplicity: delete the default cube, NumPad7, add cylinder, rotate X 90, then add torus with ext/int 1/0.75 and raise it above the cylinder. It should look like this:.  What I want is to deform the torus against the cylinder, but make it as if it's a projection. This means that, when looked from above, it will look like a normal, undeformed torus (volume changes), but when viewed with NumPad1, it will look like a round-end tube following the curvature of the cylinder, while still having the same round cross-section. To make it worse, my object needs to be deformed by a more complex surface, such as sin(x).  If I'd use soft body, it will look shrinked when looking from above, i.e. it will preserve the volume. If I use shrinkwrap, it either does a projection with its circumference intact, but all flat, or it will project it (similar to what I want), but only the bottom side vertices, like a curtain.  Mesh modifier gets me nowhere since I have to manually tweak the surrounding mesh, and curve modifier also preserves the length/volume, while also squishing the section of the torus at the bending points.  My question(s): is there a way to project (mold) the torus onto the cylinder while preserving it's cross-section? Can this be extended for more complex meshes to be projected or to be projected on?    This is, roughly, what the side view (along Y axis in the 1st pic) would look like after the deforming of the torus:    The unselected shape above is the torus (as it is in the first picture), and the selected shape is what it would look like after the deform, or molding, whatever you want to call it.  And here is what it would look like when seen from above (top view, along Z axis)): i.stack.imgur.com/HVOCo.png (can't add more than 2 pics)  The view would be the same before and after the torus is deformed. single email limit exceeded salesforce very frequently..? my email limit is exceeded after sending 10-12 emails..what could be the problem..as i have seen the limit is 1000 emails/day.i am not able to find out what could be the problem..Can some give me some suggestions. As i also know we can send 10 emails in a single transaction.I am sending emails in a batch class.  I am calling this batch class on button click and filling emailList with emailsId's.                      String query='Select id from Lead';//No use of this query                     Batch_send_emails bse=new Batch_send_emails();                     bse.query=query;                                                         bse.resendEmails=emailList;                     database.executebatch(bse);           **Batch Class**           if(!resendEmails.isEmpty())           {               for(String s:resendEmails)                {                   emailsLeads.add(s);                }           }           Messaging.SingleEmailMessage mail = new Messaging.SingleEmailMessage();         mail.setToAddresses(emailsLeads);        // mail.setCcAddresses(ccAddresses);         String subject='GoToMeeting Invitation Details for '+subject;         String body='test';          mail.setHtmlBody(body);          mail.setSubject(subject);         Messaging.sendEmail(new Messaging.SingleEmailMessage[] { mail });          } Undo apt-get remove open* ubuntu 12.04 Why must $\mathbb{Q}[\sqrt{2}, \sqrt{3}]$ -- the set of all polynomials in $\sqrt{2}$ and $\sqrt{3}$ with rational coefficients  -- contain multiplicative inverses?   I have gathered that every element of $\mathbb{Q}[\sqrt{2}, \sqrt{3}]$ takes form $a + b\sqrt{2} + c\sqrt{3} + d\sqrt{2}\sqrt{3}$ for some $a,b,c,d \in \mathbb{Q}$, and I have shown that $\mathbb{Q}[\sqrt{2}]$ and $\mathbb{Q}[\sqrt{3}]$ are both fields, but it's not clear to me why this allows for general multiplicative inverses to exist in $\mathbb{Q}[\sqrt{2}, \sqrt{3}]$. facing difficulty to create dynamic xpath for City field of UTest website I am beginner in automation field, I am trying to automated one of the module, I have tried many different XPATH to fetch proper XPATH for City field o UTest site. Im trying to automate registration module as one of my assignment. I have tried my different xpath example below:  //xpath="//body/div[@class='pac-container pac-logo']/div/span[2]/span" //div[@class='pac-item']/span[3] //div[@class='pac-item']/span //div[@class='pac-container pac-logo']/div/span[2]/../span[3]   the following code written:  //xpath to send data in field @FindBy(xpath="//input[@placeholder='Enter a location']") WebElement city;  //xpath for list to appear @FindBy(xpath="div[@class='pac-item']/span") List&lt;WebElement&gt; City;  public void EnterSignupTwo(String CiTy) throws Exception {     city.clear();     city.sendKeys(CiTy);     Thread.sleep(3000);     System.out.println("the size of List " +City.size());//printing list size      for(int i=0; i&lt;City.size(); i++ ) {         String cities = City.get(i).getText(); //storing list element in String         System.out.println("Converted value of city :"+cities); //print of total list      if(cities.contains(CiTy)) //if parameterized match with Available then click     {         System.out.println("enter specific value of city :"+City.get(i).getText());         City.get(i).click();         break;     } }   kindly refer the How to remove all files and subdirectories in a directory WITHOUT deleting the directory in bash? Is there a command to remove all files and subdirectories in a directory without deleting the directory?  For example if I have directory dontDeleteMe with subdirectories 1, 2, 3 and each subdirectory has a few pictures in it, how can I remove the subdirectories 1, 2, and 3 and all the files in the them, without removing the parent directory dontDeleteMe? I see that Stack&nbsp;Overflow is very Linux specific. Doesn't Unix &amp; Linux and Ask&nbsp;Ubuntu cover much of the same ground? I see that Unix &amp; Linux and Server&nbsp;Fault also cater to many Linux server questions. Similarly, Ask&nbsp;Ubuntu and Super&nbsp;User seems to contain most like questions. How to change all the keys of a hash by a new set of given keys Who is the eponymous Jedi in Return of the Jedi? So basically , we all know the direction outside the magnet its from NORTH to SOUTH. but what i don't understand is that what is the direction inside is it the same ?? list transparency across non consecutive slides Tensor product of a sheaf of module and a residue field Let $S$ be a scheme over a finite field $\mathbb{F}_q$ and $\mathcal{L}$ an invertible sheaf on $S$.   In Pink's article (, see section 1.4 on page 13), he states that:  For any section $\ell\in \mathcal{L}(S)$ and any point $s\in S$ we let $\ell(s)\in \mathcal{L}\otimes_{\mathcal{O}_S}k(s)$ denote the value of $\ell$ over the residue field $k(s)$ of $s$.  My question is: what is the definition of $\mathcal{L}\otimes_{\mathcal{O}_S}k(s)$?  We know we can define the tensor product of two $A$-modules or of two $\mathcal{O}_X$-modules, but what is the tensor product of an $\mathcal{O}_X$-module and a field? Include prior knowledge in regression model I've a classical dataset with real attributes and I want to perform a regression.  But, not all the entries in the training dataset are trustworthy; there is an attribute that I can turn into a confidence value (between 0 and 1) to say how much I can trust the label value. For example, if the confidence value is 0, I mustn't take the entry into consideration.  Is there a way to do this?