proof that the sum of digits of natural number are divisible by 3 iff the number is Im trying to prove that every natural number is divisble by three if and only if the sum of its digits are divisible by three.  First i proved by induction that $10^n-1$ is divisible by 9 (and therefore 3) so i could use this in the next step, would this be valid...  ANy natural number can be given a decimal representation  Let $s\in N$  Let $k_0,k_1...k_n\in ${${0,...10}$}  $S=k_0 +10k_1+100k_2+...k_n10^n$  $= 9k_1 + 99k_2 + ... + k_n(10^n-1) + (k_0+k_1+...k_n)$  All terms are divisible by 3 but the sum of the digits in S. Therefore Dividing S by 3 will leave the same remainder as dividing the digits bys 3.  Is this a valid proof of the claim? Thank you for your time
Why $9$ & $11$ are special in divisibility tests using decimal digit sums? (casting out nines & elevens) I don't know if this is a well-known fact, but I have observed that every number, no matter how large, that is equally divided by $9$, will equal $9$ if you add all the numbers it is made from until there is $1$ digit. A quick example of what I mean:  $9*99 = 891$ $8+9+1 = 18$ $1+8 = 9$  This works even with really long numbers like $4376331$ Why is that? This doesn't work with any other number. Similarly for $11$ and alternating digits sums.
I studied propositional logic, and everyday I see applications of what I learned on the internet, in mathematical books and miscelaneous resources.        One particular case is sentences in the form $a \rightarrow b$ , that is, material implications. While I fully understand the mathematical logic approach to the study of those kinds of formulas, sometimes I get confused about what someone ( using implication sentences in the ordinary language ) is trying to say.        In ordinary language, I often see affirmations in the form of an implication.  One example would be someone affirming that "the moon is made of cheese" $\rightarrow 2+2=4$. [ Kleene, Mathematical Logic ].      I perfectly understand that if we choose the interpretation that gives the antecedent as false ( and the consequent as true ) we get that the whole implication is also true, but I would like to understand EXACTLY what he is trying to say when he AFFIRMS a sentence in the form of an implication. Is he saying that under the commonly agreed interpretation of that sentence ( antecedent = false, consequent = true ), that implication becomes true?      Another example, totally different, would be someone affirming that $x \in \mathbb{N} \rightarrow x \in \mathbb{N}$ or $x \in \mathbb{Z}_{&lt;0}$. Is he now affirming the same kind of sentence or saying something totally different?      When he affirms that sentence, is he saying that he finds the sentence to be a tautology?   Why didn't he say that?          Then, with these two examples we have an affirmation of the same kind of sentence (implication) meaning completely different things, in one case he affirms to say that under the commonly agreed interpretation of the sentence, it is true. In another case, he affirms to say that there's no interpretation that makes the sentence false.       Am I missing something? There is really that ambiguity in ordinary language, when people are affirming sentences?  Can't I just take some general instance of the affirmation of a sentence in the form $a \rightarrow b$ in ordinary language and know exactly what he's trying to say, regardless of the propositions he chooses the substitute $a$ and $b$ for?       Thanks a bunch. What happens if you are refused entry between two countries? Last month, I was travelling on a backpacking trip across Russia and the Baltic states (Schengen zone). I'm an Indian by nationality. So I had to get the respective single entry visa for Russia and Schengen Europe.  At the border crossing into Estonia by road (at Ivangorod-Narva), I exited the Russian border with an exit stamp (so now I can't go back to Russia since I had a single entry visa). When I was at the immigration checkpoint of Estonia, I was heavily questioned on the intent of my travel to Europe while entering Estonia by road.   Though I finally got into Europe, It got me wondering - what happens if a foreign national is denied entry into a country not at the airport but by land between 2 countries he cannot go into ?  Generally, if you are denied entry at the airport the airline has the obligation to fly you back to your country. But what happens when you are stuck between two countries and you cant get into either ? What happens to you then ? Very specific paragraph shape Fully transparent windows in Pygame? Is it possible to get a fully transparent window in Pygame (see the desktop through it)?  I've found how to create a window without a frame, but there doesn't seem to be any obvious way to make it transparent.  I'd be willing to tie into system-specific technology/frameworks as long as there are solutions for both Windows and Mac OS X, but I'm not sure which direction to be looking.  The only topic I was able to find , which isn't something I can do for this particular project (needs to be Pygame). Integral $\int_{1}^{2011} \frac{\sqrt{x}}{\sqrt{2012 - x} + \sqrt{x}}dx$ Evaluate:  $$\int_{1}^{2011} \frac{\sqrt{x}}{\sqrt{2012 - x} + \sqrt{x}}dx$$  Using real methods only.  I am not sure what to do.  I tried finding a power series, which was too ugly.  I just need some hints, not an answer to do this integral, this is from the MIT Integration bee 2012. Question is :     Prove that the polynomial $p(x)=x^4-4x^2+8x+2$ is irreducible over the quadratic field $F=\mathbb{Q}(\sqrt{-2})$. [Hint : first use the method of proposition $11$ for the U.F.D $\mathbb{Z}[\sqrt{-2}]$(cf.Exercise $8$, Section $8.1$) to show that if $\alpha \in \mathbb{Z}[\sqrt{-2}]$ is a root of $p(x)$ then $\alpha$ is a divisor of $2$ in $\mathbb{Z}[\sqrt{-2}]$ . Conclude that $\alpha$ must be $\pm 1,\pm \sqrt{-2}$ or $\pm 2$ and hence show that $p(x)$ has no linear factor over $F$. Show similarly that $p(x)$ is not the product of quadratics with coefficients in $F$.]   What I have done so far is :  Suppose $\alpha \in\mathbb{Z}[\sqrt{-2}]$ is a root of $p(x)=x^4-4x^2+8x+2$ we would then have :  $p(\alpha)=\alpha^4-4\alpha^2+8\alpha+2=0\Rightarrow 2=\alpha(-\alpha^3+4\alpha-8)$  i.e., $\alpha$ is a divisor of $2$ in $\mathbb{Z}[\sqrt{-2}]$.  so, I have used the method of proposition $11$ for the U.F.D $\mathbb{Z}[\sqrt{-2}]$(cf.Exercise $8$, Section $8.1$) to show that if $\alpha \in \mathbb{Z}[\sqrt{-2}]$ is a root of $p(x)$ then $\alpha$ is a divisor of $2$ in $\mathbb{Z}[\sqrt{-2}]$  I do not understand why does he mentioned that $\mathbb{Z}[\sqrt{-2}]$ is U.F.D and all... I do not use that at all... It is unnecessarily confusing me or i am unnecessarily getting confused.. hint is actually misleading me :(  Now, I have to prove that $\alpha$ must be $\pm 1,\pm \sqrt{-2}$ or $\pm 2$  i.e., suppose I have $2=ab$ in $\mathbb{Z}[\sqrt{-2}]$ then, $N(2)=N(ab)\Rightarrow 4=N(a)N(b)$   i.e.,$N(a)=1\text{ or }2\text{ or } 4$ i.e., $p^2+2q^2=1\text{ or }2\text{ or } 4$ for $a=p+\sqrt{-2}q$   $p^2+2q^2=1\Rightarrow p=\pm 1 \Rightarrow a=\pm 1$ $p^2+2q^2=2\Rightarrow q=\pm 1\Rightarrow a=\pm\sqrt{-2}$  $p^2+2q^2=4\Rightarrow p=\pm 2\Rightarrow a=\pm 2$    Once I prove that those are the only divisors then I would consider :    $p(1)=(1)^4-4(1)^2+8(1)+2\neq 0$ $p(-1)=(-1)^4-4(-1)^2+8(-1)+2\neq 0$ $p(\sqrt{-2})=(\sqrt{-2})^4-4(\sqrt{-2})^2+8(\sqrt{-2})+2\neq 0$ $p(-\sqrt{-2})=(-\sqrt{-2})^4-4(-\sqrt{-2})^2+8(-\sqrt{-2})+2\neq 0$ $p(2)=(2)^4-4(2)^2+8(2)+2\neq 0$ $p(-2)=(-2)^4-4(-2)^2+8(-2)+2\neq 0$   So, no divisor of $2$ is a root..  Thus $p(x)$ do not have a root in $\mathbb{Z}[\sqrt{-2}]$  suppose I have something like :  $$x^4-4x^2+8x+2=(x^2+ax+b)(x^2+cx+d)=x^4+(a+c)x^3+(b+d+ac)x^2+(ad+bc)x+bd$$  But then, $bd=2\Rightarrow \text { b,d  are  a divisors of 2 in $\mathbb{Z}\sqrt{-2}$}$  But then we have seen that only divisors of $2$ in $\mathbb{Z}\sqrt{-2}$ are $\pm 1,\pm \sqrt{-2}$ or $\pm 2$  So, only possibilities are   $x^4-4x^2+8x+2=(x^2+ax\pm 1)(x^2+cx\mp 2)$ $x^4-4x^2+8x+2=(x^2+ax\pm \sqrt{-2})(x^2+cx\mp \sqrt{-2})$   But these are not possible...  This only tells that $p(x)$ is irreducible in $\mathbb{Z}[\sqrt{-2}]$ but then how do i show this is irreducible in $\mathbb{Q}(\sqrt{-2})$ I was expecting gauss lemma to help but it only works for integers and rationals...  So, please help me to clear this..  Thank you...   P.S : Proposition $11$ is Rational root theorem and Exercise $8$ is that ring of integers of $\mathbb{Q}(\sqrt{-2})$ is an Euclidean domain. Drawing visibility polygons in Unity (for vision cones with occlusion) I'm currently trying to create an effect like the one displayed below in Unity:    However, it seems that I can't do this with any of Unity's built-in light objects, and I simply cannot find anything online that talks about this. Is there a way to achieve this in Unity? 2D or 3D solutions are fine, as I'm sure that they're both very similar to each other, How to do something to each file in a directory with a batch script How do you iterate over each file in a directory with a .bat or .cmd file?  For simplicity please provide an answer that just echoes the filename or file path. Real valued analytic function defined on a connected set is constant Let $G$ be a connected set and $f : G \rightarrow \mathbb{C}$ a real valued analytic function. Prove that $f$ is constant.    My idea to prove the result is to prove a subset $A \neq \varnothing$ of the connected set $G$ is both open and closed. So $G=A$ Take $f(w) = a$ $$A = \{z\colon z \in G, f(z) = a\}$$ Now I want to show that $A$ is infinite. How to do it? After that it is easy to prove $A=G$. Do I need a transit visa for Malaysia? I am an Indian passport holder travelling from China to India via Kuala Lumpur, on Air Asia tickets bought separately. The layover is 4 hours and 30 minutes.  Do I need a transit visa? Which are safe and unsafe regions in Iran? Now there's a lot of propaganda in the west that leads us to believe Iran is dangerous and travellers shouldn't go there.  Obviously there are major political disagreements between several governments and the Iranian government and this should be taken into account.  But at a human level I have met many Iranians travelling elsewhere in the world and have found them to be friendly, charming, and hospitable. I have also met several western travellers who spent time in Iran, including hitchhiking around the country, who only have wonderful things to say about the country and its people.  As a traveller I've learned to trust the most what I hear from other travellers about a destination, especially ones which prefer similar ways of travel to myself. I've also learned to trust the opinions of inhabitants of countries bordering on the destination country the least.  Yet some people still insist that Iran is not safe for a traveller, or that .  I can look up for myself the general travel advisories issued by governments and disclaimers in WikiVoyage. So please don't regurgitate those an answers.  I want to know from travellers who've been there, from Iranians, or from references to specific incidents targeting foreigners/westers about which parts of Iran are no problem, and which parts should be avoided, even by experienced intrepid travellers.  (I realize any advice won't apply to Israeli citizens unfortunately. I'm Australian.) How can I change the class of an HTML element in response to an onclick or any other events using JavaScript? When running sudo apt-get update I get these following errors:  Err archive.ubuntu.com quantal InRelease Err archive.ubuntu.com quantal-updates InRelease Err archive.ubuntu.com quantal-backports InRelease Err archive.ubuntu.com quantal-security InRelease Err archive.ubuntu.com quantal Release.gpg   Unable to connect to archive.ubuntu.com:http: [IP: 91.189.92.177 80] Err archive.ubuntu.com quantal-updates Release.gpg   Unable to connect to archive.ubuntu.com:http: [IP: 91.189.92.177 80] Err archive.ubuntu.com quantal-backports Release.gpg   Unable to connect to archive.ubuntu.com:http: [IP: 91.189.92.177 80] Err archive.ubuntu.com quantal-security Release.gpg   Unable to connect to archive.ubuntu.com:http: [IP: 91.189.92.177 80] Reading package lists... Done W: Failed to fetch archive.ubuntu.com/ubuntu/dists/quantal/InRelease   W: Failed to fetch archive.ubuntu.com/ubuntu/dists/quantal-updates/InRelease   W: Failed to fetch archive.ubuntu.com/ubuntu/dists/quantal-backports/InRelease   W: Failed to fetch archive.ubuntu.com/ubuntu/dists/quantal-security/InRelease   W: Failed to fetch archive.ubuntu.com/ubuntu/dists/quantal/Release.gpg  Unable to connect to archive.ubuntu.com:http: [IP: 91.189.92.177 80] W: Failed to fetch archive.ubuntu.com/ubuntu/dists/quantal-updates/Release.gpg  Unable to connect to archive.ubuntu.com:http: [IP: 91.189.92.177 80] W: Failed to fetch archive.ubuntu.com/ubuntu/dists/quantal-backports/Release.gpg  Unable to connect to archive.ubuntu.com:http: [IP: 91.189.92.177 80] W: Failed to fetch archive.ubuntu.com/ubuntu/dists/quantal-security/Release.gpg  Unable to connect to archive.ubuntu.com:http: [IP: 91.189.92.177 80] W: Some index files failed to download. They have been ignored, or old ones used instead.   I am using Ubuntu 12.04. This is the command I used after installing Ubuntu from window installer. I am sure my laptop is connected to internet. What can I do now? How to custom (split in two)? Ibidem Early propagation of carries in the BLAKE hash function Maximum Champion Growth size I was reading about Lulu (new champion) and saw this:     Wild Growth (Ultimate) - Lulu enlarges an ally, knocking enemies away   from them and granting them a large amount of bonus health. For the   next few seconds, that ally gains an aura that slows nearby enemies.   Then I start wondering: what is the max size that a champion reach ? Is there a Maximum size ?  Example: Cho'Gath's Feast +  +  + Wild Growth + Something that make champions grow in size. Cho'Gath will grow every time ? or it will stop at some point ? Conditional expectation in AR(1) process Suppose we have a stationary AR(1) process: $Y_{t+1}=a+ \rho Y_{t} + \epsilon_{t+1}$ where $\epsilon_{t+1}$ is white noise with probability density function $\phi(.)$.   Now say we have a equation $P_{t+1}=u\cdot (A_{m}-Y_{t+1})^{+}$ where $(Y_{m}-Y_{t+1})^{+}$ = $ \max \{(A_{m}-Y_{t+1}),0\}$ and $u$ is a constant.  So $P_{t+1}=u\cdot(A_{m}-a-\rho Y_{t}-\epsilon_{t+1})^{+}$.   Can we write $E[P_{t+1} \mid Y_{t}]= u\cdot \int_{-\infty}^{(A_{m}-a - \rho Y_{t})}(A_{m}-a-\rho Y_{t}-\epsilon_{t+1})\phi(\epsilon_{t+1})d\epsilon_{t+1}$ ? Fibonacci Sequence and series limits 1) Let $F_1 = 1$, $F_2 = 1$, and $F_n = F_{n-1} + F_{n-2}$ for $n \geq 3$.   How do you prove that  $$\sum_{n=2}^\infty \frac1{F_{n-1} F_{n+1}} = 1$$   $$\sum_{n=2}^\infty \frac{F_n}{F_{n-1} F_{n+1}} = 1$$   Do you even use the formula $n(n+1)/2$?   I tried using $F_{n-1} = F_n - F_{n-2}$ and $F_{n+1} = F_n + F_{n-1}$ and then trying to figure out how to substitute those, but I've been going around in circles. How would you solve this? What tips can you give me to solve this? Is there any website that teaches this? I tried searching google and youtube but none were helpful.   2) Let $s_x = \tan(2/x)$. Where does this become nonincreasing? While using monotonic limits to establish there is a limit.   I was guessing you need to use upperbound and lowerbound but doesn't it go to infinity? Website home page design when there are multiple audiences I am in the process of creating a website which has two distinct audiences - in my case consumers and retailers.  When I first started the process just a few weeks ago I had followed the then model used by PayPal who too have to deal with distinct audiences.    However, having Googled around a bit for what the accepted wisdom is when dealing with website landing pages for segmented markets it appears to me that the current belief is that one just directs the visitor away from the landing page to other, segment focused, pages ASAP.  I have to admit that I am having trouble buying that philosophy.  To my mind it makes sense to provide at least some basic information - key benefits - for each segment on the landing page and then let viewers who linger long enough the opportunity to easily drill down to their own segment-focused page.  My own reasoning is that most first time visitors are liable to want to bounce away quickly and making them find and click on the segment page link is more likely to make them bounce than if they find enough to hold them on the landing page for a bit longer.  I'd like to have other views on this.  What are the pros and cons of the    Drive traffic to segment-focused page ASAP vs Provide some basic information - at least key benefits - for each segment on the landing page itself How can I stop mobs from despawning? I am using Spigot, some plugins, and I want to make like a zoo on a map, with mobs behind glass cubes. The problem is that when I load my map, every time some of the mobs just vanish, empty cages. I put eggs again, they spawn, I go to my other map in the multiverse (/mv tp my_other_map) and when I come back, empty cages again, always the same ones...   I've  read it's a question of distance of me (distance in cubes unit), if I'm too far away, the mobs will despawn... Is there a way for them to stay?