I have an arithmetic sequence, $a+dn$, where $a$ and $d$ are constants and $n$ is the term number. How would I efficiently calculate solutions to the following equation, $$ (a+dn)\%b=0 $$ where $b$ is also a constant. (and $\%$ means modulo) I attempted the question myself and found out that I really just need to find the smallest $n$ that satisfies the equation and find the rest by adding ${kb\over GCD(d,b)}$ to $n$, where $k$ is any integer. I also noticed that the smallest solution will always be less than $b$ and if no solution exist within this range, then there is no solution. So I tried to find the solution by brute forcing from 0 to $b$, until I got a solution. But this method is not nearly as efficient as I need it to be.
Problem :   Solve in $\mathbb N$ :   $8n+9\equiv 0\pmod{1163}$  The mathematics give me : $n=435+1163k$ , $k\in\mathbb N$   $1163$ is a prime number   $8n\equiv -9\pmod{1163}$   $8n\equiv 1154\pmod{1163}$   I don't know any ideas to complete my work ?   I have to see your hints
How to prove that if $\lim_{n \rightarrow \infty}a_n=A$, then $\lim_{n \rightarrow \infty}\frac{a_1+...+a_n}{n}=A$ I'm wondering if I have a sufficient proof of the following:   If $(a_n)$ is a sequence such that $\lim_{n \rightarrow \infty}a_n=A$, then $\lim_{n \rightarrow \infty}\frac{a_1+...+a_n}{n}=A$.   My approach:  For all $\varepsilon &gt; 0$, there exists $N$ such that for all $k&gt;N$, $|a_k -A|&lt;\varepsilon$. So we can break the limit up as follows $$\lim_{n\rightarrow \infty} \frac{a_1+...+a_k}{n} + \lim_{n \rightarrow \infty}\frac{a_{k+1}+...+a_n}{n} \overset{\epsilon \rightarrow 0}{=} 0 + \lim_{n \rightarrow \infty}\frac{nA}{n}=A$$  Is this on the right track, or am I missing something about breaking up the limit in the way I have? How to choose a good research problem? I am working to get a PhD, maybe next year this time around. Currently, this is my second year. I have published 3 papers. All of them are not about one problem but consider different problems. My strategy to find a research idea/problem is given below.  1 - Read papers whose titles are either interesting or I have some apriori understanding of those titles.  2 - Understand those papers as best as I can. That is, I try to understand each detail presented in the paper except the simulation results (this is because I think simulation results are hard to reproduce).  3 - I analyze the paper and try to find a very small problem which can be addressed (i.e. I can solve it) by the methodology of the paper.  4 - If I am successful at step 3 I start working on that problem otherwise I repeat the process from step 1.  By applying this strategy I have been able to write 3 papers and I have submitted a fourth (I am not satisfied with those papers because at very best they can be considered average papers). But my concern is that the problems addressed in those papers do not belong to one area. As far as I have seen, for the profiles of some of the renowned researchers in my field their PhD work is focused on one area. One main thing I console myself with is that almost all of those researchers had good supervisors. My supervisor does not even read the draft/reviews of my papers and the suggestions that he provides during the meetings are so broad that the problem becomes intractable. Under these circumstances what are your opinions: Is there any better strategy to do research?   I will be very thankful to you for providing me suggestions or commenting on my dilemma. Is there a way of installing Ubuntu (Windows Subsystem for Linux) on Win10 (v1709) without using the Store? The Microsoft Store app is disabled by Group Policy by my university. I have administrator rights on my computer and I was wondering if there was a way of installing Windows Subsystem for Linux (WSL) directly? Or maybe via Chocolatey or some other command line method? I would like to download and install the Ubuntu  on windows without using the Microsoft store.  Note that since version 1709 (Fall Creator's Upgrade), only enabling Developer Mode is no longer sufficient. Sci Fi action movie where an elite group of people control business through a game they play online Unique (non-repeating) random numbers in O(1)? Why does an atom remain uncharged after emission of an alpha particle? When an alpha particle is emitted, two protons and two neutrons leave the nucleus but the electrons remain the same in number. Why does the atom remain uncharged although it appears it should have a net charge of $-2e$? Prove the equality: $\det\left[\begin{smallmatrix} -2a &a+b &a+c \\ b+a&-2b &b+c \\ c+a&c+b &-2c \end{smallmatrix}\right] = 4(a+b)(b+c)(c+a)$ i have to prove:   $$\begin{vmatrix} -2a &amp;a+b  &amp;a+c \\   b+a&amp;-2b  &amp;b+c \\   c+a&amp;c+b  &amp;-2c  \end{vmatrix} = 4(a+b)(b+c)(c+a)$$   I have tried many calculations between the rows and columns of the determinant to get to the answer i want to solve the exercise however none of them gave me something right. Can someone help? Selecting pixels in raster using polygon selection tool? I want to reclassify landcover image manually with respect to online imagery. I am looking for a software/plugin/script to manually select raster pixels using polygon selection tool and could be able to enter new value.  I have Arcgis with spatial analyst license. I need to prove that the language $\mathscr L=\{0^n \big|\text{n is the square of a natural number}\}$ is non regular using the pumping lemma   My try:  $\mathscr L=\{\overbrace{\epsilon}^{0^2},\overbrace{0}^{1^2},\overbrace{0000}^{2^2},\overbrace{000000000}^{3^2},\overbrace{0000000000000000}^{4^2},\dots\}$  Suppose that $\mathscr L$ is regular so $\exists$ a word '$x$' with length of at least $n$   $|x|\geq n $ such that  $(1)\,\,\,|uv|\leq n$  $(2)\,\,\,|v|\geq 1$  $(3)\,\,\,uv^iw \in \mathscr L$  Now, let as choose the word $\color{blue}{x=0^{n^2}}$  $|x|\geq n$ so we can use $(1)-(3)$  $x=\underbrace{000000\dots}_{n^2\text{ times}}$  $x=uv0^kw=\underbrace{uv0^k}_{\text{n zeros   }}\overbrace{w}^{n^2-n\text{ zeros }}$  Let as choose $\color{blue}{i=??}$      I'm difficult choosing the word '$\color{blue}x$' and choosing  $\color{blue}i$ to come to contradiction with the lemma The Frobenius Coin Problem Prove that the expression cannot be a power of 2 I am a bit financially strapped and received a Gateway GT5056 tower. The tower has an AMD dual core 64-bit processor in it. Will Ubuntu 12.04 run on this machine? I will not have any other application other than LibreOffice installed. Answers to a  mention that it is illegal for a US citizen to enter/leave their home country on a foreign passport. However this raises the following questions:   What is the maximum prescribed penalty for violating these rules? Are US immigration officials known for actually applying the penalty? E.g. the Cuba embargo banning visits to Cuba was almost never enforced, so it's possible this law is ignored as well. If $f$ is measurable and $fg$ is in $L^1$ for all $g \in L^q$, must $f \in L^p$? I installed Windows 7, which ate Ubuntu's boot file. When starting up the computer, it now goes straight to Windows, without giving me the option of booting Ubuntu.   How can I get Ubuntu back? I would like to understand how Physicists think of space-time in the context of String Theory. I understand that there are $3$ large space dimensions, a time dimension, and $6$ or $7$ (or $22$) extra dimensions, and all these dimensions need to fit together in a way such that the extra dimensions are compactified (with a Calabi-Yau or $G_2$ structure).   My question, however is not about the possible $10$, $11$ or $26$ dimensional manifolds that may be possible, but about whether string theorists consider space-time as somehow quantized (or discrete), or rather as a continuous manifold, or are both options possible? In other words, can strings move continuously through space, or are there a discrete set of locations where strings can be, and does string theory rule out one of the options?  How about the same question in loop quantum gravity (LQG)? Should I think of the spin networks in LQG as describing a discrete space-time?  Thanks for your insight, or any references you may be able to provide! subequations and array in braces I need something like this:  \left\{ first line  (1a) second line (1b) third line  (1c) \right.   all lines should be aligned to left Can't mount an IDE HDD - "can't read superblock" error message I'm trying to mount my HDD,  /dev/sdb, which is connected via an enclosure but every time I try do so I get an error message   mount: /dev/sdb: can't read superblock   How do I solve this problem and mount my HDD? I used the jQuery Form plugin for asynchronous form submission. For forms that contain files, it copies the form to a hidden iframe, submits it, and copies back the iframe's contents. The problem is that I can't figure out how to find what HTTP status code was returned by the server. For example, if the server returns 404, the data from the iframe will be copied as normal and treated as a regular response.  I've tried poking around in the iframe objects looking for some sort of status_code attribute, but haven't been able to find anything like that.    The $.ajax() function can't be used, because it does not support uploading files. The only way to asynchronously upload files that I know of is using the hidden iframe method. Web applications that want to force a resource to be downloaded rather than directly rendered in a Web browser issue a Content-Disposition header in the HTTP response of the form:  Content-Disposition: attachment; filename=FILENAME  The filename parameter can be used to suggest a name for the file into which the resource is downloaded by the browser.  (Content-Disposition), however, states in  (The Filename Parameter) that the file name can only use US-ASCII characters:     Current [RFC 2045] grammar restricts   parameter values (and hence   Content-Disposition filenames) to   US-ASCII.  We recognize the great   desirability of allowing arbitrary   character sets in filenames, but it is   beyond the scope of this document to   define the necessary mechanisms.   There is empirical evidence, nevertheless, that most popular Web browsers today seem to permit non-US-ASCII characters yet (for the lack of a standard) disagree on the encoding scheme and character set specification of the file name. Question is then, what are the various schemes and encodings employed by the popular browsers if the file name “naïvefile” (without quotes and where the third letter is U+00EF) needed to be encoded into the Content-Disposition header?  For the purpose of this question, popular browsers being:   Firefox  Internet Explorer  Safari Google Chrome  Opera