$A$ is a set of integers and $\lvert A \rvert = 501$.   For any $a \in A$, $1 \le a \le 1000$.  Prove that there are integers $a,b \in A$, such that $a \vert b$. (a divides b)  The question doesn't specify it, but I assume that $a \neq  b$.  (Hint: Note that every integer $x$ can be uniquely presented as $x = y · 2 ^k$ where $k &gt; 0$ and $y$ is odd. Consider the function $x \to y$)  Can I get some help?
This is exercise 1.6.1 of   Suppose that 501 distinct integers are selected from 1 . . . 1000. Show that there are distinct selected integers $a$ and $b$ such that $a | b$. Show that this is not always true if 500 integers are selected.  This seems like it should be simple but it's has been stumping me for a few days.  Here's my idea for a proof:  Let $S$ be a set of 500 distinct integers between 1 and 1000 that don't divide each other. Let $S_{1-500}$ and $S_{501-1000}$ partition $S$ into sets containing the elements ≤ and > than 500, respectively. Then consider $2 S_{1-500}$ and $\frac{1}{2}S_{501-1000}$, the result of multiplying the elements of these sets by 2 and 1/2, respectively. Because elements of S are coprime, $S_{1-500}$, $S_{501-1000}$, $2 S_{1-500}$, and $\frac{1}{2}S_{501-1000}$, are disjoint. Also note that each of these sets is a subset of the integers from 1 to 1000. One can deduce that the cardinality of the union of these sets is 1000, and therefore they form a partition of the integers from 1 to 1000.  Now, I know this demonstrates that you can't add another number to $S$ that is coprime with all the elements in $S$, which means that a coprime subset of the integers from 1 to 1000 can have at most 500 elements. Can someone articulate why this is true for me? Also, this was in a section on the Pigeonhole principle, can someone rework the proof to utilize that, and possibly make it simpler? Thanks.
When I create a form using the form API, the resulting html includes all sorts of unwanted divs. For example, if I use:  $form = array(     'myFieldset' =&gt; array(         '#type' =&gt; 'fieldset'     ) );   The resulting html looks like this:   &lt;form action="/drupal-7.15/?q=chercheur" method="post" id="form-chercheur" accept-charset="UTF-8"&gt;     &lt;div&gt;         &lt;fieldset class="form-wrapper" id="edit-myfieldset"&gt;             &lt;div class="fieldset-wrapper"&gt;             &lt;/div&gt;         &lt;/fieldset&gt;         &lt;input type="hidden" name="form_build_id" value="form-87Dv5ezVpaLI5mD345ofMkLgPhsv_dawudUp73ePQTc" /&gt;         &lt;input type="hidden" name="form_token" value="sXXINWr7rmParm0tRJoEb6QKGIH_QH-ffRSOtnvxYY0" /&gt;         &lt;input type="hidden" name="form_id" value="form_Chercheur" /&gt;     &lt;/div&gt; &lt;/form&gt;   As you can see, two extraneous divs are added here. They are making the javascript I'm writing uselessly complicated, so I was wondering if there was a way to remove the extraneous divs. Ideally, I would like to obtain something like this:  &lt;form action="/drupal-7.15/?q=chercheur" method="post" id="form-chercheur" accept-charset="UTF-8"&gt;     &lt;fieldset class="form-wrapper" id="edit-myfieldset"&gt;     &lt;/fieldset&gt;     &lt;input type="hidden" name="form_build_id" value="form-87Dv5ezVpaLI5mD345ofMkLgPhsv_dawudUp73ePQTc" /&gt;     &lt;input type="hidden" name="form_token" value="sXXINWr7rmParm0tRJoEb6QKGIH_QH-ffRSOtnvxYY0" /&gt;     &lt;input type="hidden" name="form_id" value="form_Chercheur" /&gt; &lt;/form&gt;   Is this possible? I am looking at  (dual low-voltage power amplifier). On the fourth page, there is a sample circuit; right at the V+ input there is a symbol that looks like a capacitor. However, it looks like it is shaded in. What is that? (I'm aware of  but it has no answers…)  Let be $u_0 \in \mathbb{R}$ and the sequence $(u_n)_n$ defined by: $u_{n + 1} = \frac12 \arctan(u_n)$.  I define also: $v_n = 2^n u_n$, so I can show that: $\lim (u_n)_n = 0$ (by studying $x \mapsto \frac12 \arctan(x)$), thus, I can show that $(v_n)_n$ is monotone and converges because it is bound.  Now, I conclude: $u_n \sim \dfrac{l}{2^n}$, I'd like to determine $l$ more precisely.  Here is what I tried, I suspect $l$ to be something like $f(\pi)$ for some $f$ :   push the asymptotic expansion of $\arctan$ to the 2nd order and reinject it ; use $\arctan(u_n) + \arctan(1/u_n) = \dfrac{\pi}{2}$ ; use series techniques to look for $\sum v_{n + 1} - v_n$, maybe conclude using Cesaro summation How can I compile this successfully? I have to use algpseudocode package to improve my paper look like more professional   But I can't compile successfully.  \usepackage{amsmath} \usepackage{algorithm} \usepackage[noend]{algpseudocode}  \makeatletter \def\BState{\State\hskip-\ALG@thistlm} \makeatother  \begin{document} \begin{algorithm} \caption{AIP}\label{euclid} \begin{algorithmic}[1] \Procedure{MyProcedure}{} \State $\textit{locationAgent} \gets \text{position of }\textit{{agent}}$ \State $\textit{locationMD} \gets \text{position of }\textit{{MD}}$ \If {\textit{distance(eNB,$agent_i$)}}&gt;$\delta$ \BState \emph{loop}: \State $j \gets \textit{patlen}$ \If {$\textit{string}(i) = \textit{path}(j)$}\Return TURE \State $j \gets j-1$. \State $i \gets i-1$. \State \textbf{goto} \emph{loop}. \State \textbf{close}; \EndIf \State $i \gets i+\max(\textit{delta}_1(\textit{string}(i)),\textit{delta}_2(j))$. \State \textbf{goto} loop \EndProcedure Pointer to class data member "::*" Solve a system of equations with Euclid's Algorithm Speak Slower or Speak Slowlier? AFAIK the correct grammar for "speak slow" is "speak slowly" (slowly being an adverb). Please correct me if I am mistaken.  But in daily life I have not heard anyone saying "Speak slowlier".  I think I heard folks saying "Speak slower" but I also think it is incorrect.  Which is the best or correct form to convey the message?  Could that be the longish "Speak more slowly?" something like wye-delta and delta-wye connection.  we can originally know wye-delta connection is step-down transformer and second side voltage phase is being to -30 degree. And also delta-wye connection is step-up transformer and second side voltage phase is being to +30 degree.  But the other side, we also can make wye-delta connection is step-up transformer if it get turn ratio like 1:2. So in this case maybe will be become +30 degree in second side voltage.   apply this principle in scott connection, how can work it?   case1) turn ration 1:2 - step up second side voltage's angle +90 degree?  case2) turn ration 2:1 - step down second side voltage's angle -90 degree?  if you have any information or homepage link, can you leave it for me?  thank you so much. Java String split is not working Modified pattern does not see the pattern color option Recently I noticed that many people post .htaccess files here with:  &lt;IfModule mod_rewrite.c&gt;   Sometimes this even appears several times in the file! It obviously checks to see if mod_write is in fact enabled but the statements it protects are necesary to those sites anyway. So if the IfModule fails, the website is broken anyway.  Is there an advantage in having the &lt;IfModule mod_rewrite.c&gt; fail rather than the RewriteRule statements below it?  mod_write now is so essential and ubiquitous that I do not check for it anymore. Might this cause a vulnerability of some kind? This is a question that's been nagging me for some time. I always thought that C++ should have been designed so that the delete operator (without brackets) works even with the new[] operator.  In my opinion, writing this:  int* p = new int;   should be equivalent to allocating an array of 1 element:  int* p = new int[1];   If this was true, the delete operator could always be deleting arrays, and we wouldn't need the delete[] operator.  Is there any reason why the delete[] operator was introduced in C++?  The only reason I can think of is that allocating arrays has a small memory footprint (you have to store the array size somewhere), so that distinguishing delete vs delete[] was a small memory optimization. Can I apply for two Schengen visas if I have two subsequent trips within a short period? I am traveling to Germany on May 1st for an exhibition for which I have already obtained a visa and its validity is up to May 13th. I am returning on May 12th and then traveling on May 17th to UK, and from UK to Switzerland for vacation with family. As time is very short for applying for a Schengen visa again after returning on the 12th with May 13-14 being a Saturday and Sunday, can I apply for Schengen visa again and can I get 2 valid Schengen visa? Xdotool using "DISPLAY=:0" not works in Crontab How do I iterate over a range of numbers defined by variables in Bash? How do I iterate over a range of numbers in Bash when the range is given by a variable?  I know I can do this (called "sequence expression" in the Bash ):   for i in {1..5}; do echo $i; done   Which gives:     1    2    3    4    5   Yet, how can I replace either of the range endpoints with a variable? This doesn't work:  END=5 for i in {1..$END}; do echo $i; done   Which prints:     {1..5} Why is it difficult to differentiate between  and ? Is it because we don't clearly understand how both of these phenomenon takes place?   My thoughts: From an answer to one of my earlier question  I could understand that light waves behave differently when it strikes a slit which is of several times bigger than its wavelength and when it strikes a slit of size comparable to it its wavelength.In the former case the different points on the slit acts as wave-front sources producing circular wave fronts which interfere to form the diffraction pattern but in the later case plane wave-front striking a slit after passing through it forms circular wave-fronts and two such wave-fronts from two different slits interfere to form the interference pattern. But, I'm not satisfied by that because I don't get any intuition about points on the slit acting as wavefront sources. So, what is actually happening? Multiple outputs for beamer handouts I'm currently generating some presentations with beamer for students of my course. Some like to print the material, so I created a handout. Some students prefer to have 2,3,4,6 etc. pages per sheet of A4 paper. How can I go about to automatize the following code, so that it outputs 4 different files in 1x2,1x3,2x2,2x3 format?  I'd like to avoid having to compile it again everytime for the different formats.  \documentclass[a4paper]{article} \usepackage{pdfpages}  \begin{document} \includepdf[pages=1-last,nup=1x2,landscape=false,frame=true,             noautoscale=false,scale=1,delta=0mm 5mm]{Pres7Chap13-Handout.pdf} \end{document} I wanted to know how I could configure multi-touch gestures in Ubuntu. In windows I can do this using the synaptics driver software.  I have already used synclient to configure the taps, palmdetect, scroll, etc. But I'd also want gesture support. Is is even possible?  System-Info: Ubuntu 12.10, Synaptics clickpad Forward vs Forwards When does the set enter set theory? I wonder about the foundations of set theory and my question can be stated in some related forms:   If we base Zermelo–Fraenkel set theory on first order logic, does that mean first order logic is not allowed to contain the notion of sets? The axioms of Zermelo–Fraenkel set theory seem to already expect the notion of a set to be defined. Is there are pre-definition of what we are dealing with? And where? In set theory, if a function is defined as a set using tuples, why or how does first order logic and the axioms of Zermelo–Fraenkel set theory contain parameter dependend properties $\psi(u_1,u_2,q,...)$, which basically are functions?