The Number of different $n \times n$ skew-symmetric matrices with each element being equal to either $0$ or $1$ or $-1$ where $n=5$ is ?
The no. of all possible matrices of order $3\times 3$ with each entry $0$ and $1$ and $ -1$.
Prove $\gcd(a+b,a^2+b^2)$ is $1$ or $2$ if $\gcd(a,b) = 1$ Assuming that $\gcd(a,b) = 1$, prove that $\gcd(a+b,a^2+b^2) = 1$ or $2$.   I tried this problem and ended up with $$d\mid 2a^2,\quad d\mid 2b^2$$ where $d = \gcd(a+b,a^2+b^2)$, but then I am stuck; by these two conclusions how can I conclude $d=1$ or $2$?  And also is there any other way of proving this result? Why does the question's closed status disappear when clicking to load a new edit? What is $\gcd(0,a)$, where $a$ is a positive integer? I have tried $\gcd(0,8)$ in a lot of online  gcd (or hcf) calculators, but some say $\gcd(0,8)=0$, some other gives $\gcd(0,8)=8$ and some others give $\gcd(0,8)=1$. So really which one of these is correct and why there are different conventions? Apparently recurrences like this cannot be solved with the Master Theorem:  $T(n) = 2T\left(\frac{n}{2}\right) + \frac{n}{\log(n)}$  Because $n^{\log_b(a)} = n^1$ is not a polynomial multiple of $f(n) = \frac{n}{\log(n)}$ since $\frac{n}{\frac{n}{\log(n)}} = \log(n)$, which is less than $n^\epsilon$ for any positive $\epsilon$.  But when is this really even an issue?   For example why wouldn't the time complexity be $n \log n$ using case 2 of the Master Theorem? Unrolling the recursion:  $$T(n) = n\sum_{k=1}^{\log_2(n)} \frac{1}{\log_2(2^k)}$$  $$T(n) = n\sum_{k=1}^{\log_2(n)} \frac{1}{k}$$  Doesn't this suggest $\Theta(n \log n)$ even though it supposedly cannot be done with Master Theorem? How could a statement be true without proof? Godel`s incompleteness theorem states that there may exist true statements which have no proofs in a formal system of particular axioms. Here I have two questions; 1) How can we say that a statement is true without a proof? 2) What has the self reference to do with this? Godel sentence "G" can say that SUB(a,a, no prove) but could be this just arbitrarily judgement about non-provability of "a" because it may simply has a proof which is not yet revealed or discovered? Show that $\mathbb{Q}(\zeta_n)$ is Galois over $\mathbb{Q}$ and $Gal(\mathbb{Q}(\zeta_n)/\mathbb{Q}))\cong \mathbb{Z_n}^*$ Question 1: For $n \in \mathbb{N}$ explain why $\mathbb{Q}(\zeta_n)$ is Galois over $\mathbb{Q}$   We must show that it is normal and separable.   $\mathbb{Q}(\zeta_n)/\mathbb{Q}$ is normal since it is a splitting field for $X^n-1 \in\mathbb{Q}[X]$ (I think... Is this correct?).  It is separable because it is of characteristic $0$  Question 2: $Gal(\mathbb{Q}(\zeta_n)/\mathbb{Q}))\cong \mathbb{Z_n}^*$  Is this because the primitive nth root of unity $\zeta_n$ acts as a generator for a cyclic group? I am unable to fill out a decent proof on this.   Thanks for your help How to show hidden folders in File Manager (Nautilus) on Ubuntu? 1.I will study hard until I get a high score on this exam. 2.I will study hard until I've gotten a high score on this exam.  I think these sentences are grammatically correct. but if I add a specific time, I think these sentences have to be only this one:  3.I will study hard until I get a high score on this exam on July 25th.  And how about these?  a. I'd like to watch all of the holocaust movies present in the world until I've become an expert on it. b. I'd like to watch all of the holocaust movies present in the world until I become an expert on it.  Since there's no a word representing a specific time, both of them are okay, right? And what I'd like to know is the difference between &quot;have gotten&quot; and &quot;get&quot;. What I know about them is we can use &quot;future perfect&quot; to refer to unspecific future event that might happen before some accident. but &quot;get&quot; refers to future event that could be either a specific scheduled event or unspecific future event. that's why if I add a specific time mentioning, I have to use &quot;get&quot;. What software is used to generate the reputation graphs on Stack Exchange? I'm particularly interested in the zooming functionality which is done very well. Were these graphs generated using off the shelf software or were they coded from scratch? how to change a pose lib using script I want to make a button that changes the active poselib to another one from the list, a specific one. I look in the console and can't see any operator. any ideas? How do I get a consistent byte representation of strings in C# without manually specifying an encoding? How do I convert a string to a byte[] in .NET (C#) without manually specifying a specific encoding?  I'm going to encrypt the string. I can encrypt it without converting, but I'd still like to know why encoding comes to play here.  Also, why should encoding even be taken into consideration? Can't I simply get what bytes the string has been stored in? Why is there a dependency on character encodings? How to prove the inequality $2\sqrt{n + 1} − 2 \le 1 +\frac 1 {\sqrt 2}+\frac 1 {\sqrt 3}+ \dots +\frac 1 {\sqrt n} \le 2\sqrt n − 1$? Prove that for any positive integer $n$, $$2\sqrt{n + 1} − 2 \le 1 +\frac 1 {\sqrt 2}+\frac 1 {\sqrt 3}+ \dots +\frac 1 {\sqrt n} \le 2\sqrt n − 1$$ Progress I think Riemann sum should be used for the middle term. I got the limit as $n\to \infty$ of the function inside using Riemann sum but what I cannot get are the lower and upper bound. htdigest file format Getting different answers when integrating using different techniques Question:  Is it possible to get multiple correct results when evaluating an indefinite integral?  If I use two different techniques to evaluate an integral, and I get two different answers, have I necessarily done something wrong?    Often, an indefinite integral can be evaluated using different techniques.  For example, an integrand might be simplified via partial fractions or other algebraic techniques before integration, or it might be amenable to a clever substitution.  These techniques give different results.  For example, looking over a few other questions on MSE:      From : evaluate $$ \int x(x^2+2)^4\,\mathrm{d}x. $$    Via the substitution $u = x^2+2$, this becomes $$ \int x(x^2+2)^4\,\mathrm{d}x       = \frac{1}{10}x^{10} + x^8 + 4x^6 + 8x^4 + 8x^2 + \frac{32}{5} + C.     $$ However, multiplying out the polynomial and integrating using the power rule gives $$ \int x(x^2+2)^4\,\mathrm{d}x       = \frac{1}{10}x^{10} + x^8 + 4x^6 + 8x^4 + 8x^2 + C    $$     From :  evaluate   $$ \int \frac{1-x}{(x+1)^2} \,\mathrm{d}x. $$    Simplifying the integrand using partial fractions then integrating gives $$ \int \frac{1-x}{(x+1)^2} \,\mathrm{d}x       = \frac{-2}{(x+1)} - \ln|x+1| + C.    $$ Via integration by parts, we get $$ \int \frac{1-x}{(x+1)^2} \,\mathrm{d}x       = \frac{x-1}{(x+1)} - \ln|x+1| + C.    $$     From : evaluate   $$ \int \frac {\tan(\pi x)\sec^2(\pi x)}2\,\mathrm{d}x. $$    Using the substitution $u = \sec(\pi x)$, this becomes $$ \int \frac {\tan(\pi x)\sec^2(\pi x)}2\,\mathrm{d}x       = \frac {\sec^2(\pi x)}{4\pi} + C.    $$ Using the substitution $u = \tan(\pi x)$, this becomes $$ \int \frac {\tan(\pi x)\sec^2(\pi x)}2\,\mathrm{d}x       = \frac {\tan^2(\pi x)}{4\pi} + C.    $$ Idiom for a speech with redundant information I've been thinking about switching to Ubuntu full time, but there are still some programs left behind that I haven't found any equivalent to yet. Does anyone know of a music player similar to Foobar2000 (I've read it doesn't handle well in WINE)? What is a good Photoshop book for photographers? I am interested in not too basic techniques (which I can more or less figure out by myself), which on the other hand give good results, without the need to be an expert pro. In counting degrees of freedom of a linear molecule, why is rotation about the axis not counted? How do I remember which units counter which in StarCraft 2? Render an InputText based on Selected Option In my VF page I have an apex:selectList with a List of Vendors (e.g. Alcatel,Cisco ec..), the last option is "Other".   When the user select "Other" an apex:inputText should be rendered  where he can writes the Vendor. I'm not able to render the inputText Field   My VF page:  &lt;apex:outputPanel id="vendor2"&gt; &lt;p&gt;&lt;b&gt; Vendor: &lt;/b&gt;&lt;/p&gt; &lt;apex:selectList value="{!vendor}" multiselect="False" size="1"&gt;     &lt;apex:actionSupport event="onchange" action="{!specifyVendor}" rerender="otherPanel"/&gt;     &lt;apex:selectOptions value="{!vendors}"/&gt; &lt;/apex:selectList&gt;&lt;p/&gt; &lt;/apex:outputPanel&gt; &lt;br/&gt; &lt;apex:outputPanel id="otherPanel"&gt;     &lt;apex:inputText value="{!otherVendor}" rendered="{!vendor=='Other'}"&gt; Other: &lt;/apex:inputText&gt; &lt;/apex:outputPanel&gt;     &lt;br/&gt;&lt;br/&gt;    The custom controller:      public with sharing class ExcelPageController {          public String vendor {get; set;}//Vendor Name         public String otherVendor {get; set;}//Vendor Name          public List&lt;SelectOption&gt; getVendors(){             System.debug('getVendors');                 List&lt;SelectOption&gt; options = new List&lt;SelectOption&gt;();             options.add(new SelectOption('Cisco','Cisco'));             options.add(new SelectOption('Alcatel','Alcatel'));             options.add(new SelectOption('Tibco','Tibco'));             options.add(new SelectOption('Other','Other'));             return options;         }     public PageReference specifyVendor(){         System.debug(vendor);         return null;     } }