Show that if N is relatively prime to 10, then there exists a multiple that consists only of 1s.
prove that there are infinitely many numbers of the form $x = 111....1$ such that $31\mid x$
Solving linear congruence (modular inverse or fraction) via gcd Bezout equation Isn't finding the inverse of $a$, that is, $a'$ in $aa'\equiv1\pmod{m}$ equivalent to solving the diophantine equation $aa'-mb=1$, where the unknowns are $a'$ and $b$? I have seem some answers on this site (where the extended Euclidean Algorithm is mentioned mainly) as well as looked up some books but there is no mention of this. Am I going wrong somewhere or is this a correct method of finding modular inverses? Also can't we find the Bézout's coefficients by solving the corresponding diophantine equation instead of using the extended Euclidean Algorithm? scanf not taking in data Let $R$ be a commutative Noetherian ring (with unity), and let $I$ be an ideal of $R$ such that $R/I \cong R$. Then is it true that $I=(0)$ ?  I know that a surjective ring endomorphism of a Noetherian ring is also injective, and since there is a natural surjection from $R$ onto $R/I$ we get a surjection from $R$ onto $R$, but the problem is I can not determine the map explicitly and I am not sure about the statement. Please help. Thanks in advance. Exporting coordinates from LAS files to CSV? Is there any tool available for extracting the X, Y and Z coordinates from a .las file and exporting them to a .csv or similar file? Can sudo be reinstalled after being removed? This seems to be a chicken-egg problem. The most common task using sudo is installing and removing software.  sudo apt-get purge &lt;appname&gt;   But sudo itself can be removed.  sudo apt-get purge sudo # Do not run this command on production computers!   This is where the fun comes  ubuntu@ubuntu:~$ sudo bash: /usr/bin/sudo: No such file or directory   Although it's obvious that no person in his right mind will purge sudo (other than me), someone can be fooled to run this command (not directly, in its hex mode, or whatever it's called) or a person could SSH in disguised as tech guru and do the mess.  So is there a way of reinstalling sudo? How to display Trash and Home folder icons? How to correct for small p-value due to very large sample size I am running into a problem where an independent variable, which should have no predictive power on the dependent variable based on domain knowledge, comes out with very small p-value because the sample size is very large(~100,000). If I only use &lt; 5000 data points, then the p-value becomes large enough to support the prior that the variable is insignificant. However, I don't think tweaking the sample size to get to the desired conclusion is a good practice. Is there any procedure to adjust for small p-value simply due to huge sample size? I am taking an abstract algebra course and I am really interested about the topic.  So much so that I spend most of my time reading supplementary materials.  I consequently know a lot more theorems than the ones covered in class.    In a recent quiz, I used a theorem in one of my proofs that was not mentioned in the lecture since.  My professor gave me partial marks for my answer for the reason that we didn't cover this theorem in class even though the proof was completely valid!  I can't really see where he is coming from.  How would someone be able to use a theorem correctly if he doesn't know its proof?  I was quite baffled by his comment.   Do you agree with this?  EDIT: The professor just Emailed me and said that after thinking for a while, he decided to award me the full mark for the question. He also mentioned that he didn't want discourage me from studying the subject (since he saw most of my previous quiz grades were full marks) and I was passionate about it, but kindly requested that I include any proof for a theorem that I cite from now on in his exams. He also sent a broadcast message to the whole class indicating this. Confused about AMD/Intel When I go to the downloads page to get Ubuntu, it automatically downloads the AMD version, but my computer has no hardware with AMD on it. I only have Intel and Nvidia. Also, when I install this AMD version on my computer, it works (other than the constant random freezing of the computer when I use Ubuntu). I don't really understand why the amd version works on a computer with an intel processor. Number of $n^2\times n^2$ permutation matrices with a 1 in each $n\times n$ subgrid I found the following question in a paper I was trying to solve:  The following figure shows a $3^2 \times 3^2$ grid divided into $3^2$ subgrids of size $3 \times 3$. This grid has $81$ cells, $9$ in each subgrid.    Now consider an $n^2 \times n^2$ grid divided into $n^2$ subgrids of size $n \times n$. Find the number of ways in which you can select $n^2$ cells from this grid such that there is exactly one cell coming from each subgrid, one from each row and one from each column.  My try:  Since we have $n^2$ rows, $n^2$ columns and $n^2$ subgrids in total, we have to choose one and only one cell from each of them. Let's choose them one at a time. We can choose the first cell in $n^4$ many ways. Then, we'll have to avoid that subgrid, that column and that row that we've chosen the first one from when choosing the second cell. So, we have $n^4-n^2-2n(n-1)$ choices. We can continue this to get the total number of possible ways. But, I think there's a hole. Say, we've chosen the first cell from the subgrid of the up-left corner and the second from the subgrid just right to it so that it doesn't violate any rules. Then, when finding the number of ways we can choose the third cell, we would have substracted some of the cells twice. I think you get it. Please, if anyone can help me solving this problem, it'd be greatly appreciated. "Z-level" in TikZ Quotient rings of Gaussian integers I used this isomorphism today but now I'm having trouble justifying it. The norm function isn't additive so I can't come up with a ring isomorphism to prove the following:     For any $\,a+bi\in\Bbb Z[i],\,\gcd(a,b)=1$, we have a ring isomorphism   $$\Bbb Z[i]/\langle a+bi\rangle\,\cong\Bbb Z/(a^2+b^2)\Bbb Z=:\Bbb Z_{a^2+b^2}.$$   Could someone show me an isomorphism between these rings to prove this? Search favorite questions by tag Favorite questions on Stack Overflow I feel is highly unused feature of all the features provided on the site. (This again is my personal opinion)  I have favorited around 90 question till date and it gets very very very difficult if I want to search for a question which I remember I have favorited, which defeats the entire purpose of one favouriting a question, so here is my small suggestion.  Favorites by tags  Screen 1        Screen 2        Please share your opinions on this. Manipulating Partial Derivatives of Inverse Function In lectures we're told:  $$\dfrac {\partial y} {\partial x} = \dfrac 1 {\dfrac {\partial x} {\partial y}}$$  as long as the same variables are being held constant in each partial derivative.  The course is 'applied maths', i.e non-rigorous, so don't confuse me.  But anyway, if we have:  $$\xi = x - y \qquad \eta = x$$  Then $\dfrac {\partial x} {\partial \xi} = 0$ and $\dfrac {\partial \xi} {\partial x} = 1$.  The rule presumably fails because one of the partial derivatives is $0$.  But then isn't this rule useless? Since I cannot use it without checking that the partial derivative isn't in fact $0$, but then I've worked out the partial derivative manually anyway. Cross-site AJAX requests I need to make an AJAX request from a website to a REST web service hosted in another domain.  Although this is works just fine in Internet Explorer, other browsers such as Mozilla and Google Chrome impose far stricter security restrictions, which prohibit cross-site AJAX requests.  The problem is that I have no control over the domain nor the web server where the site is hosted. This means that my REST web service must run somewhere else, and I can't put in place any redirection mechanism.  Here is the JavaScript code that makes the asynchronous call:  var serviceUrl = "http://myservicedomain"; var payload = "&lt;myRequest&gt;&lt;content&gt;Some content&lt;/content&gt;&lt;/myRequest&gt;"; var request = new XMLHttpRequest(); request.open("POST", serviceUrl, true); // &lt;-- This fails in Mozilla Firefox amongst other browsers request.setRequestHeader("Content-type", "text/xml"); request.send(payload);   How can I have this work in other browsers beside Internet Explorer? Do the electric and magnetic components of an electromagnetic wave really generate each other? Frequently when EM waves are taught, it is said that the change in electric field causes a change in the magnetic field, which then causes a change in the electric field, and so on and so forth.   But from my understanding of basic electromagnetics, it is not necessarily a changing electric field that creates a change in magnetic field, but instead an accelerating charge. and it is this charge that creates both the electric and magnetic field. Again, from my understanding, a changing magnetic field is not generated by a changing electric field, but instead just happens to always be present perpendicular to a changing electric field due to the laws of electromagnetism.   Am I wrong? Or is the "mutual generation" concept between the electric and magnetic components of an EM wave an actual thing? What is your simplest explanation of string theory? Apex Array Slice Sometimes I like to use a Queueable to perform work on a list of Ids or sObjects over the heaver Batchable Interface.   However, to do this, we need a very solid implementation of an Array slice method.  Most other programming languages have a native slice method, but Apex does not.  While I have written my own version of slice and it's working fine, I would be interested in hearing about other implementations of slice - the more efficient the better.  My version is in this Utilities class on github   It's modelled after the Javascript version described on the mozilla site   I would expect that the usage and output be similar to this:  String[] arr = ['zero', 'one', 'two', 'three']; String[] sliced = Utilities.slice(arr,1, 3);  System.debug(arr);      // ['zero', 'one', 'two', 'three'] System.debug(sliced);   // ['one', 'two'] How to uninstall chrome OS and install Ubuntu 16.04 fully (not Crouton) on a Chromebook I want to uninstall Chrome OS and install Ubuntu instead. How would I do this? Autorun.inf in Windows 8/8.1 for USB Drives I have created a file named autorun.inf to make one of my software setup run when the usb is plugged into Windows 8/8.1 OS .  This is my autorun.inf code:   [AutoRun] shellexecute="Mem.exe //It would be nice if i could add path to this like my Doc//mem.exe UseAutoPlay=1   Is there any software that let's us to create auto-run files for USB Sticks?  Is there any way to add new options in right click menu only on my USB window?