Prove: $a\equiv b\pmod{n} \implies \gcd(a,n)=\gcd(b,n)$
Show that if $a \equiv b \pmod n$, $\gcd(a,n)=\gcd(b,n)$
The equals() method in Java works unexpectedly on Long data type Changing CSS Values with Javascript jQuery - auto size text input (not textarea!) Adapt V-brakes bicycle to disk brakes I've been thinking in changing my v-brakes to disc brakes. Now that my rear v-brake can't brake anymore I think that it is time change it. I've already seen . My frame is prepared for disc and, as I don't want to change my brake lever, I'll go with mechanical brakes (does  look ok?);  The only problem is the rear wheel hub that has not the six holes for the disk. Is it ok if I just change the rear wheel hub and keep the front one? Package dbus is not configured yet state that "at best" is an idiom. But, what is the grammatical function of "at best" (for example, in the below sentences?)        Their response to the proposal was, at best, cool.   The government's response seems to have been at best confused and at   worst dishonest.   If he drops the course now, at best he’ll get an incomplete, and he could fail.      Thanks Unable to mount NTFS external hard drive Inline elements shifting when made bold on hover I created a horizontal menu using a HTML lists and CSS. Everything works as it should except when you hover over the links. You see, I created a bold hover state for the links, and now the menu links shift because of the bold size difference.  I encounter the same problem as . However, the post does not have proper solution. I've searched everywhere for a solution and can't find one. Surely I can't be the only one trying to do this.  Does anyone have any ideas?  P.S: I don't know the width of the text in menu items so I cannot just set the width of the li items.    .nav { margin: 0; padding: 0; } .nav li {      list-style: none;      display: inline;      border-left: #ffffff 1px solid;  } .nav li a:link, .nav li a:visited {      text-decoration: none;      color: #000;      margin-left: 8px;      margin-right: 5px;  } .nav li a:hover{      text-decoration: none;      font-weight: bold;  } .nav li.first { border: none; } &lt;ul class="nav"&gt;     &lt;li class="first"&gt;&lt;a href="#"&gt;item 0&lt;/a&gt;&lt;/li&gt;     &lt;li&gt;&lt;a href="#"&gt;item 1&lt;/a&gt;&lt;/li&gt;     &lt;li&gt;&lt;a href="#"&gt;item 2&lt;/a&gt;&lt;/li&gt;     &lt;li&gt;&lt;a href="#"&gt;item 3&lt;/a&gt;&lt;/li&gt;     &lt;li&gt;&lt;a href="#"&gt;item 4&lt;/a&gt;&lt;/li&gt; &lt;/ul&gt; I have been trying around different ranges of summations:  $$ \sum_{(m,n) \in \mathbb{Z}^2} \frac{m^2 - n^2}{(m^2 + n^2)^2} = 0$$  That's not any good.  What about if we restrict to $m, n \in \mathbb{Z}$ as positive integers.    $$ \sum_{m &gt; 0, n &gt; 0} \frac{m^2 - n^2}{(m^2 + n^2)^2} = 0$$  Now here's an anti-symmetry.  I do not like taking out the symmetry, but perhaps I can ask about:   $$ \sum_{m &gt;  n &gt; 0} \frac{m^2 - n^2}{(m^2 + n^2)^2}  = \; ? \tag{$*$} $$  There doesn't seem to be a change of variables that can work.  And we've used symmetry about as much as we can.  This looks related to:  $$  \sum \frac{1}{n^2} = \frac{\pi^2}{6}$$  Perhaps this other series ($*$) also has a special value. Horizontal line of custom length like an — Is there a way to get a custom-length horizontal line that appears like a "longer em-dash", if you will? I know how much space is required to install Ubuntu.  How much space does a Ubuntu installation take up on the disk? Fix formatting errors of page number > 99 in table of contents? This is a follow up question of .  The formatting problem has nothing to do with cleanthesis.sty.  The problem is that if I use the charter font, page numbers greater than 99 are not formatted correctly in the table of content.  \documentclass[draft=on]{scrbook} \usepackage{blindtext} \usepackage{charter}  \listfiles  \begin{document} \tableofcontents \section{Section 1} \blindtext[100] \section{Section 2} \blindtext[500] \section{Section 3} \end{document} If $p$ is congruent to $2 \pmod 3$, how can I prove that all $a$, $1 \le a \le p-1$ are cubic residues $\mod p$? If $p$ is congruent to $2 \pmod 3$, how can I prove that all $a$, $1 \le a \le p-1$ are cubic residues $\mod p$?  Here's what I've done:  $1^3$ congruent to $1 \pmod p$ thus, 1 is a cubic residue, Also $(p-1)^3=(p-1) \pmod p$ implies  $$(p-1)^3 - (p-1) = (p-1)[(p-1)^2 -1] = (p-1)(p^2 -2p) = p(p-1)(p-2) $$ implies $$p|(p-1)^3 -(p-1)$$ Thus  $p-1$ is a cubic residue $\mod p$. Now I don't know how to show for $1\lt a \lt p-1$ are all cubic residues when $p$ is congruent to $2\pmod 3$. How to access specific raw data on disk from java "Geometric" or "Geometrical"? I have read the excellent answers to  - my question is specifically about usage.   Is there a best practice for deciding between the variants geometric and geometrical in writing? A  shows that geometrical was more popular until about 1965, when they switched. What grammatical rules, if any, would be relevant here? For what powers $k$ is the polynomial $n^k-1$ divisible by $(n-1)^2$? What is the difference between fastboot oem unlock and fastboot oem unlock-go? I'm having issues unlocking my Nexus 7 2013 WiFi's bootloader, and it was suggested that I try fastboot oem unlock-go instead of fastboot oem unlock. Both seem to act the same on my device, however both also fail on my device so it is possible that the difference appears after the failure on my end comes. Regardless, I would like to know: What is the difference between these two commands? I'm not managing to find documentation. Enumerate alignment problem in theorem environment I'm using amsart. When I create an enumeration inside a theorem environment, the numbering is not aligned properly:  Theorem 14. (1) (2) (3) (4)   How do I shift the (2),(3),... to be aligned with (1)?  MWE:  \documentclass{amsart} \newtheorem{thm}{Theorem}[section]  \begin{document} \begin{thm} \begin{enumerate} \item Item 1 \item Item 2 \item Item 3 \end{enumerate} \end{thm} \end{document} Help Me Understand My Bicycle's Gears My book says:     For most of the small objects, both are same. But for mammoth ones, they are really different ones. And in a gravity-less environment, COG is absent; COM still exists.   Ok, what's the big deal when things are small and big? How these two :  &amp; ?