Hey I would like to show that     $a\mid b\Rightarrow \varphi(a)\mid\varphi(b)\qquad a,b\in\mathbb{N}$   where $\varphi(n)$ is the the totient function.  My try:  Let $a,b\in\mathbb{N}$ and $a\mid b$. By Eulers formula we have that $$ \varphi(a)=a\cdot \prod_{\substack{p \mid a\\ p\ \text{prime}}} \Big(1-\frac{1}{p}\Big). $$ Now since $a\mid b$ by the Fundamental Theorem of Arithmetic we can factorise $a$ and $b$ into $$ a=\prod_{i=1}^{n} p_i^{\alpha_i} \qquad b=\prod_{i=1}^{m} p_i^{\beta_i} $$ with $p_i$ being primes, $n \le m$, and $\alpha_i \le \beta_i$ for $1 \le i \le n$. Using this we can say  $$ \left.\prod_{i=1}^{n} \Big(1-\frac{1}{p_i}\Big) \middle| \prod_{i=1}^{n}\Big(1-\frac{1}{p_i}\Big)\right. $$ and this concludes the result.   Now my question is that is this correct? If it is is there another way to arrive to this conclusion? My way of describing this felt a bit clucky.  Happy new years eve to all!
Prove that if $d$ divides $n$ then $φ(d)$ divide $φ(n)$ for $φ$ denotes Euler’s $φ$-function.  I know that $d|n$ mean there exists some integer $k$ such that $n=kd$, but how can I use this to prove $φ(d)$ divide $φ(n)$
In today's pattern recognition class my professor talked about PCA, eigenvectors and eigenvalues.   I understood the mathematics of it. If I'm asked to find eigenvalues etc. I'll do it correctly like a machine. But I didn't understand it. I didn't get the purpose of it. I didn't get the feel of it.        I strongly believe in the following quote:     You do not really understand something unless you can explain it to your grandmother. -- Albert Einstein   Well, I can't explain these concepts to a layman or grandma.   Why PCA, eigenvectors &amp; eigenvalues? What was the need for these concepts? How would you explain these to a layman? Did Emperor Palpatine know about Luke and Leia from the beginning? Did Emperor Palpatine know that Padme had given birth to twins while it was happening?  Was it his long-term goal to have Vader only discover his offspring later, at some "correct" moment? Is there any way (such as a command or environmental variable) to determine what window manager is the one actively running in the current session? So I have learned in class that light can get red-shifted as it travels through space. As I understand it, space itself expands and stretches out the wavelength of the light. This results in the light having a lower frequency which equates to lowering its energy.  My question is, where does the energy of the light go? Energy must go somewhere!   Does the energy the light had before go into the mechanism that's expanding the space? I'm imagining that light is being stretched out when its being red-shifted. So would this mean that the energy is still there and that it is just spread out over more space? $\phi \times \theta$ is faithful iff $(|Z(H)| ,|Z(K)|)=1$ for faithful characters $\phi \in Irr(H)$ and $\theta \in Irr(K)$ . For $n \geq 2$, prove that $(1- \frac{1}{4})(1- \frac{1}{9})(1- \frac{1}{16})...(1- \frac{1}{n^2}) = \frac{n+1}{2n}$ For $n \geq 2$ prove that $(1- \frac{1}{4})(1- \frac{1}{9})(1- \frac{1}{16})...(1- \frac{1}{n^2}) = \frac{n+1}{2n}$  We need to use induction.   The Principle of Mathematical Induction, Theorem 4.2.1, states that $n_0 \in \mathbb{Z}$. For each integer $n \geq n_o$, let $P(n)$ be a statement about $n$. Suppose that the following two statements are true:   $P(n_o)$ $( \forall n \geq n_o)[P(n) \rightarrow P(n+1)]$   Then, for all integers $n \geq n_o$, that statement $P(n)$ is true.  Suppose,  $(1- \frac{1}{4})(1- \frac{1}{9})(1- \frac{1}{16})...(1- \frac{1}{n^2}) = \frac{n+1}{2n}$, the equation, is true for some natural number $k$.  $\frac{1}{2^2})*...*(1- \frac{1}{k^2}) = \frac{k+1}{2k}$  The induction hypothesis $P(k) \rightarrow P(k+1)$ must hold for any natural number $k, P(k),$ from the basis and inductive step.  $(\frac{1}{2^2})*...*(1- \frac{1}{k^2})*(1- \frac{1}{(k+1)^2}) = \frac{k+1}{2k}(1- \frac{1}{(k+1)^2})$  $\leftrightarrow$ $\frac{k+1}{2k}(\frac{(k+1)^2}{(k+1)^2}- \frac{1}{(k+1)^2})$  $\leftrightarrow$ $(\frac{(k+1)}{2k})( \frac{(k+1)^2-1}{(k+1)^2})$  $\leftrightarrow$ $( \frac{(k^2+2k+1-1}{(2k)(k+1)})$  $\leftrightarrow$ $( \frac{(k^2+2k}{(2k)(k+1)})$  $\leftrightarrow$ $( \frac{k+2}{(2)(k+1)})$  We need to demonstrate that   $(\frac{1}{2^2})*...*(1- \frac{1}{n^2}) = \frac{n+1}{2n}$  holds true for $n \geq 2$  The question is do I just substitute a number that is greater than two after I did the induction? How can I prove this sequence is decreasing? My problem is how to prove  $$\frac{1+2}{1}  &gt; \frac{1+3+\frac{3^2}{2!}}{1+2} &gt; \frac{1+4+\frac{4^2}{2!}+\frac{4^3}{3!}}{1+3+\frac{3^2}{2!}}&gt;\cdots $$  i am sure that the sequence is decreasing as limit exp(1).  However, the reason why the sequence is decreasing is not evident.  Thanks if you consider this issue and make me relief. How do I add swap after system installation? I did not create a swap partition during Ubuntu installation.  Later, I freed up some space and made a swap partition.  Now after each boot up, I am manually opening gparted to right-click the swap partition to turn on the 'swapon' option.  How can I automatically enable the swap partition at boot?  Also, if I had not created the swap partition, what are my other options? Continuous function positive at a point is positive in a neighborhood of that point Pretty much the problem asks if a function is continuous at the point $c$ and $f(c) &gt; 0$ then there exists a $d &gt; 0$ such that $\forall x$, $f(x) &gt; 0$ with $|x-c| &lt; d$.   I can understand what the problem means. That if a function is positive at a point then there is another point that's really close that'll also be positive. I can not prove this in a formal way though. I've tried using the intermediate value theorem and I do not know hot to implement it. Any ideas? Let $f$ and $g$ be continuous at $ x=a$, with $f(a)<g(a)$. Prove that there exists a neighborhood $I$ of a such that $f(x)<g(x)$ for all $x \in I$. Export multiple objects to .obj Is it possible for an encryption method to exist which is impossible to crack, even using quantum computers? Quantum computers are known to be able to crack  a broad range of cryptographic algorithms which were previously thought to be solvable only by resources increasing exponentially with the bit size of the key. An example for that is .  But, as far I know, not all problems fall into this category. On , we can read     Researchers have developed a computer algorithm that doesn’t solve problems but instead creates them for the purpose of evaluating quantum computers.   Can we still expect a new cryptographic algorithm which will be hard to crack using even a quantum computer? For clarity: the question refers to specifically to the design of new algorithms. So I finally killed the Ender Dragon! As I was climbing the portal back to the world, I noticed that on top of the column there was an egg. So I attacked, mined, clicked and nothing happened, it just teleported away. What does the egg do and how do I get it? Change the selected value of a drop-down list with jQuery I have a drop-down list with known values. What I'm trying to do is set the drop down list to a particular value that I know exists using jQuery.  Using regular JavaScript, I would do something like:  ddl = document.getElementById("ID of element goes here"); ddl.value = 2; // 2 being the value I want to set it too.   However, I need to do this with jQuery, because I'm using a CSS class for my selector (stupid  client ids...).  Here are a few things I've tried:  $("._statusDDL").val(2); // Doesn't find 2 as a value. $("._statusDDL").children("option").val(2) // Also failed.      How can I do it with jQuery?     Update  So as it turns out, I had it right the first time with:  $("._statusDDL").val(2);   When I put an alert just above it works fine, but when I remove the alert and let it run at full speed, I get the error     Could not set the selected property. Invalid Index   I'm not sure if it's a bug with jQuery or Internet Explorer 6 (I'm guessing Internet Explorer 6), but it's terribly annoying. Packet Tracer 7 on Ubuntu 18.04 cannot launch I am having some issues with Packet Tracer 7 on Ubuntu 18.04: when I launch it with ./PacketTracer7 within the /opt/pt/bin directory, it opens but close immediately with a lot of:     "Warning: libpng warning: iCCP: known incorrect sRGB profile"   error msgs on the console.  The thing is, I installed PacketTracer7 on Ubuntu 16.04 and I remember that I had that issue with the libqt5multimedia.so.5 dependency, but I was able to find the solution (thanks to askubuntu and google) by installing the dependencies. So, everything perfect with 16.04.  But I'm a bit lost now. I can't realize where the issue come from, because I know the qt5 libs are installed on the system.  This is what I see when I run dpkg --get-selections:  libqt4-xmlpatterns:amd64            install libqt5concurrent5:amd64             install libqt5core5a:amd64              install libqt5dbus5:amd64               install libqt5gui5:amd64                install libqt5multimedia5:amd64             install libqt5multimediaquick-p5:amd64          install libqt5multimediawidgets5:amd64          install libqt5network5:amd64                install libqt5opengl5:amd64             install libqt5opengl5-dev:amd64             install libqt5positioning5:amd64            install libqt5printsupport5:amd64           install libqt5qml5:amd64                install libqt5quick5:amd64              install libqt5quickparticles5:amd64         install libqt5quicktest5:amd64              install libqt5quickwidgets5:amd64           install libqt5script5:amd64             install libqt5scripttools5:amd64            install libqt5sensors5:amd64                install libqt5sql5:amd64                install libqt5sql5-sqlite:amd64             install libqt5svg5:amd64                install libqt5test5:amd64               install libqt5webchannel5:amd64             install libqt5webkit5:amd64             install libqt5webkit5-dev:amd64             install libqt5widgets5:amd64                install libqt5x11extras5:amd64              install libqt5xml5:amd64       I've tried to uninstall pt and the qt5 libs and install everything again, but nothing... the same issue.  Did anybody install PacketTracer7 on Ubuntu 18.04?  I would be very happy to have your help for doing it!  Thanks and forgive me for my poor english :) Is there a general formula for finding the primitive of $$e^{ax}\sin(bx)?$$  I've done this manually with $a=9$ and $b=4$ using Euler's formulas. But it takes a bit of time. Is there a pattern here? Evaluate $\lim\limits_{n\to\infty}\frac{\sum_{k=1}^n k^m}{n^{m+1}}$ Can I print a custom paper on standsard size in word? Analogue of closed graph theorem This is the analogue of closed graph theorem for compact space          Suppose that $X$ and $K$ are metric spaces, that $K$ is compact, and that the graph of $f: X \rightarrow K$ is a closed subset of $X \times K$. Prove that $f$ is continuous.      Because metric space is first countable, all we need is to prove that if $x = \lim_{n \rightarrow \infty} x_{n}$, then $f(x) = \lim_{n \rightarrow \infty} f(x_{n})$. Suppose $x = \lim_{n \rightarrow \infty} x_{n}$, then because $K$ is compact, sequence $\{f(x_{n})\}$ has a subsequence $\{f(x_{n_{i}})\}$ converge. Because the graph of $f$ is closed, we must have $\lim_{n \rightarrow \infty}\{f(x_{n_{i}})\} = f(x)$. All we need to prove now is $\{f(x_{n})\}$ converge to $f(x)$, but I can't prove that. Can any one help me. I really appreciate. Thanks How to output multiple image sizes from one design? I am designing an app for Android and as you may know that Android uses 6 different density buckets from LDPI to XXXHDPI.   Is there a way to save multiple outputs of different sizes from one design using either Photoshop, Illustrator, or another software?  If not will I have to resize it 6 times or is there an easy way?