The question pertains to the question in the title. That is,     Show that if $n$ is not prime then $\Bbb Z /n\Bbb Z$ is not a field.   I realise that this has many duplicates, but my question is rather this;     Why does it not suffice to say that if $n$ is not prime then $\lvert\Bbb Z /n\Bbb Z \rvert = n \neq p^m$ for some prime $p$ and integer $m$ and so is not a field by definition?   I realise this somewhat misses the reason why it's not a field, but would it suffice as an answer to this question?
Let $n \in ℕ$. Show that the ring $ℤ/nℤ$ is a field if and only if $n$   is prime.   Let $n$ prime. I need to show that if $\bar{a} \neq 0$ then $∃\bar b: \bar{a} \cdot \bar{b} = \bar{1}$. Any hints for this ?  Suppose $ℤ/nℤ$ is a field. Therefore: for every $\bar{a} \neq 0$ $∃\bar b: \bar{a}\cdot \bar{b}=1$. How can I show that $n$ must be prime ?
Bind application to a specific network interface I have tried ForceBindIP, but it has a significant downside - it doesn't affect the children of the application I'm trying to bind, it only affects the application itself. It also can't force an application to always run through a specified interface, it has to be run through forcebindip.exe every time. It becomes a problem with applications like League of Legends where the process tree looks like that:    The launcher runs the patcher, the patcher run the client, etc. I can only affect the parent of all these processes in the tree, so the actual game is not bound to the interface I want, making this whole venture pointless.  Is there a more modern alternative to ForceBindIP for Windows 7? There are many questions similar to this one on this site, but they are mostly old. Maybe there is now a better way to solve this problem?  My current idea is to do the following:   Set up local 3proxy server bound to the desired interface. Run the game through Proxifier or similar software configured to run through that local proxy.   I'm not sure if that will work, but even if it will, it seems like a sub-optimal solution. Do you guys have any better ideas?  Edit: My idea didn't work :(  Edit 2: Basically, what I'm trying to achieve is bind a few applications to a regular interface, while VPN is running. The reason is that I need to connect through VPN most of the time, but some applications (such as games) don't work properly this way, because of higher ping and other issues. how can we show $\frac{\pi^2}{8} = 1 + \frac1{3^2} +\frac1{5^2} + \frac1{7^2} + …$? Let $f(x) = \frac4\pi \cdot (\sin x + \frac13 \sin (3x) + \frac15 \sin (5x) + \dots)$. If for $x=\frac\pi2$, we have $$f(x) = \frac{4}{\pi} ( 1 - \frac13 +\frac15 - \frac17 + \dots) = 1$$ then obviously :  $$ 1 - \frac13 +\frac15 - \frac17 + \dots=\frac{\pi}{4}$$ Now how can we prove that: $$\frac{\pi^2}{8} = 1 + \frac1{3^2} +\frac1{5^2} + \frac1{7^2} + \dots$$ How are you dealing with the  in your machine learning models? Let's say you used to predict the number of sales each month. The virus affected your results last year and it will affect for at least a couple of months. So your model, I guess, is missing a lot. What are some approaches we can use to fix our models? What is the plural of "cottage tariff"? How to run adb and fastboot commands from Termux? Why can ECC key sizes be smaller than RSA keys for similar security? Unable to log in to Stack Exchange Can I update an early Airport Time Capsule? I have an Apple Time Capsule model A1254. It has a 1Tb drive. Is it possible to replace the drive with a larger capacity unit? It seems a shame to replace the whole unit. Thanks How to find vendor and model of unknown drivers I reinstalled windows on a new machine and the drivers are not present for the product network card.   When in device manager all I get is the question mark and I am unable to figure out the actual name of the vendor or model of the card. This is making my search to find the correct driver hard.   Anyone know how I to find out more about the hardware that does not have drivers? How can I make my Photoshop text appear like it does on the web? Do I really have to design everything in Photoshop with anti-aliasing set to "none"?  I'm sure there is a better way.    I'd like to have a setting so that when picking a basic web-safe (such as Tahoma) in Photoshop so I can see what it will actually look like when on the web.  For example: The difference between Tahoma with anti-aliasing set to "crisp" in Photoshop and how it renders in Firefox is major.  It looks nice and smooth in Photoshop and looks like crap on the web.  Is there a anti-aliasing choice that is best to start with (that has the most similar outcome) when planning to have the content live on the web? Re-mounting an un-mounted USB stick Martingale formulation of Bellman's Optimality Principle From David Williams' Probability with Martingales     Related question:   What I tried: For no 2, if $\ln Z_n - n \alpha$ is a supermartingale, then for $m &lt; n$, $$E[\ln Z_n - n \alpha | \mathscr F_m] \le \ln Z_m - m \alpha$$ $$ \to E[\ln Z_N - N \alpha | \mathscr F_0] \le \ln Z_0 - 0 \alpha$$ $$ \to E[\ln Z_N - N \alpha] \le \ln Z_0$$ $$ \to E[\ln \frac{Z_N}{Z_0}] \le N \alpha$$ Is that right?  For no 1, for $m &lt; n$, $$E[\ln Z_n - n \alpha | \mathscr F_m] = E[\ln Z_n | \mathscr F_m] - n \alpha$$ $$ = E[\ln \frac{Z_n}{Z_0} | \mathscr F_m] - n \alpha + \ln Z_0$$ $$ = E[\ln \frac{Z_n}{Z_{n-1}} | \mathscr F_m] + ... + E[\ln \frac{Z_{m+1}}{Z_{m}} | \mathscr F_m]$$ $$+ \ln \frac{Z_m}{Z_{m-1}} + ... + \ln \frac{Z_{1}}{Z_{0}} - n \alpha + \ln Z_0$$ Note that $$Z_n - Z_{n-1} \le |Z_n - Z_{n-1}| \le C_n |\epsilon_n - \epsilon_{n-1}| \le 2C_n \le 2Z_{n-1}$$ Also, I think $$\ln \frac{Z_n}{Z_{n-1}} \le |Z_n - Z_{n-1}|$$ Hence we have $$E[\ln \frac{Z_n}{Z_{n-1}} | \mathscr F_m] + ... + E[\ln \frac{Z_{m+1}}{Z_{m}} | \mathscr F_m]$$ $$+ \ln Z_m - n \alpha$$ $$\le E[2Z_{n-1} | \mathscr F_m] + ... + E[2Z_{m} | \mathscr F_m]$$ $$+ \ln Z_m - n \alpha$$ $$= 2(E[Z_{n-1} | \mathscr F_m] + ... + E[Z_{m} | \mathscr F_m])$$ $$+ \ln Z_m - n \alpha$$ One thing to do would be to show that $$2E[Z_{n-1} | \mathscr F_m] + ... + 2E[Z_{m} | \mathscr F_m] \le (n-m) \alpha$$ possibly by showing that $2E[Z_{\{\cdot\}} | \mathscr F_m] \le \alpha$ How would I do that? Something else: $$E[\ln \frac{Z_n}{Z_m} | \mathscr F_m] + \ln Z_m - n \alpha$$ $$ \le E[|Z_n - Z_m| | \mathscr F_m] + \ln Z_m - n \alpha$$ $$ \le E[2Z_m | \mathscr F_m] + \ln Z_m - n \alpha$$ $$ \le 2Z_m + \ln Z_m - n \alpha$$ Now $2Z_m \le (n-m) \alpha?$ We have $$2Z_m = 2\sum_{k=1}^{m} C_k(\epsilon_k - \epsilon_{k-1})$$ $$\le 2\sum_{k=1}^{m} Z_{k-1}$$ I'm not quite sure how to show that $$2\sum_{k=1}^{m} Z_{k-1} \le (n-m)\alpha$$ if that's even true. How can/else can I approach this problem?  For no 3, no idea. Any hints? I think it has something to do with a stopping time. Might we have to use the ff lemma (red box):    ? Why does parameter expansion with spaces without quotes work inside double brackets "[[" but not inside single brackets "["? I'm confused with using single or double brackets. Look at this code:  dir="/home/mazimi/VirtualBox VMs"  if [[ -d ${dir} ]]; then     echo "yep" fi   It works perfectly although the string contains a space. But when I change it to single bracket:  dir="/home/mazimi/VirtualBox VMs"  if [ -d ${dir} ]; then     echo "yep" fi   It says:  ./script.sh: line 5: [: /home/mazimi/VirtualBox: binary operator expected   When I change it to:  dir="/home/mazimi/VirtualBox VMs"  if [ -d "${dir}" ]; then     echo "yep" fi   It works fine. Can someone explain what is happening? When should I assign double quotes around variables like "${var}" to prevent problems caused by spaces? What is the gradient of $f(x, y, z) = \sqrt{x^2+y^2+z^2}$? What is the gradient of $f(x, y, z) = \sqrt{x^2+y^2+z^2}$?  I know that $\nabla f = \left\langle\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z}\right\rangle$ or equivalently $\nabla f = \frac{\partial f}{\partial x}\mathbf{i} + \frac{\partial f}{\partial y}\mathbf{j} + \frac{\partial f}{\partial z}\mathbf{k}$.  Eventually, I am going to evaluate this gradient at a point and then dot it with some given unit vector to find a directional derivative. I think maybe what is troubling me is the radical in $f$. Will I have to use the chain rule in order to find the three partial derivatives?  I tried this:  $$\frac{\partial f}{\partial x} = \frac{1}{2}(x^2 + y^2 + z^2)^{-\frac{1}{2}}2x$$ $$= \frac{x}{\sqrt{x^2 + y^2 + z^2}}$$  That result looks ugly to me. However, if I did that partial derivative correctly, then I think I know what I am doing. Objective: To separate text column from figures column in every page I don't know if this has been asked before. But so far nothing has come up with my search.  In some books, the figures are placed at the outer side of the page like in the picture below.    Is this achievable in TeX/LaTeX? That is how does one set up the page so that all text will appear in a 'text column' and the figures appear in a 'figures column' in every page? I have a bunch of raw data values that are dollar amounts and I want to find a confidence interval for a percentile of that data.  Is there a formula for such a confidence interval? Proof of $f^{-1}(B_{1}\setminus B_{2}) = f^{-1}(B_{1})\setminus f^{-1}(B_{2})$ I want to prove the following equation:  $$ f^{-1}(B_{1}\setminus B_{2}) = f^{-1}(B_{1})\setminus f^{-1}(B_{2}) $$  Is this a valid proof? I am not sure, because at one point I am looking at $f(x) \in B_1$, but then $x \in f^{-1}(B_1)$ could be actually some different points.  $$\begin{align*} x \in f^{-1}(B_{1}\setminus B_{2}) &amp;\iff f(x) \in B_{1}\setminus B_{2} \\ &amp;\iff f(x) \in B_{1} \land f(x) \notin B_{2} \\ &amp;\iff x \in f^{-1}(B_{1}) \land x \notin f^{-1}(B_{2}) \\ &amp;\iff x \in f^{-1}(B_{1})\setminus f^{-1}(B_{2}) \end{align*}$$ has a weird case of the mono, while .  Also, that question is missing some content at the end: count() returns a different amount than a simple select How to find normal and lognormal moments, given partial information? $Y=\ln(X)$. $X$ is lognormal and $Y$ is normal. If all I know is the arithmetic mean of $Y$ and the standard deviation of $X$. What is the formula to calculate the arithmetic mean of $X$ and the standard deviation of $Y$?