I have no idea how to do this problem; please consider helping me:     If $3|(a^2 + b^2)$, show that $3|a$ and $3|b$.
After a lot of messing around today I curiously observed that $a^2+b^2$ is only divisible by 3 when both $a$ and $b$ contain factors of 3.  I am trying to prove it without using modular arithmetic (because that would be way too easy), but finding it very difficult to do so.  Is there an easy way to prove this without using modular arithmetic?  I am also interested in a more general statement.  Namely, I want to find the values of $Z$ for which $Z \mid \left( a^2+b^2 \right)$ necessarily implies that $Z \mid \gcd \left( a,b\right)$.  We know that $Z$ cannot be 5, because $3^2+4^2=5^2$.  More generally, if $Z$ is the largest element of a pythagorean triple then the above implication does not hold.
In JavaScript, the "this" operator can refer to different things under different scenarios.   Typically in a method within a JavaScript "object", it refers to the current object.  But when used as a callback, it becomes a reference to the calling object.  I have found that this causes problems in code, because if you use a method within a JavaScript "object" as a callback function you can't tell whether "this" refers to the current "object" or whether "this" refers to the calling object.  Can someone clarify usage and best practices regarding how to get around this problem?     function TestObject() {             TestObject.prototype.firstMethod = function(){                       this.callback();                       YAHOO.util.Connect.asyncRequest(method, uri, callBack);              }              TestObject.prototype.callBack = function(o){               // do something with "this"               //when method is called directly, "this" resolves to the current object               //when invoked by the asyncRequest callback, "this" is not the current object               //what design patterns can make this consistent?               this.secondMethod();             }             TestObject.prototype.secondMethod = function() {              alert('test');             }         } I have a few questions regarding CAN bus load  Is there any derived equation/formula to calculate the bus load?  I need transmit below frames(Assume Standard frame format) at baud rate 500 kbps on the bus 5ms  ----- 50 frames. 10ms ----- 10 frames. 100ms ---- 30 frmaes. Event messages ---- 10 frames.   How to calculate bus load, does anyone knows kindly illustrate the same?  What is the maximum bus load on CAN bus? Why does the QGIS Server WMS layer not load in QGIS Desktop? For me at least,  has answers that are not sorted by score:    Is this normal?  I've tried refreshing the page a few times, but the order doesn't change... My final goal is to generate Impulse Response Functions in R.  I have variables that are non stationary when I set k = 5 in a Unit Root test, and they are cointegrated which to my understanding prompts the use of the VECM, from which the Vec2Var argument is used to then generate IRFs. However, my response functions from this methodology do not decay over time and mostly do not revert to the zero line.  Furthermore, I noticed that when I input first differenced variables into the VECM as opposed to level data above, response function do revert to the zero line. Therefore my first question is: 1) is it appropriate to use differenced variables in a VECM model?  Secondly, as an alternative I am considering restricting the number of lags used in the Unit Root tests so that the variables are not all non-stationary; in this case 2) would the use of VECM be void and the VAR with first differences be a more appropriate model? Again, using differenced data is giving me better response functions that revert to zero in the long run. However, 3) is it ok to use the VAR for stationary/non-stationary data in levels where these variables are still cointegrated?  Also, I am using growth rate variables, 4) should I still long transform all variables and use logs in all tests?  Thank you for the help! Find $\lim_{x \to 0} (\frac{\tan(x)}{x})^{\frac{1}{x}}$ Find $$\lim_{x \to 0} (\frac{\tan(x)}{x})^{\frac{1}{x}}$$   My first idea to solve this was to try to evaluate it and then apply the L'Hospital's rule. This is what I managed to achieve: $$\left(\frac{\tan(x)}{x}\right)^\frac{1}{x}=e^{\frac{1}{x}\ln\left(\frac {\tan(x)}x \right)} $$ However, this is problematic, because the L'Hospital's rule is not applicable in the exponent, therefore my transformation was not a good shot. What transformation should I apply to get the desired form? Correct usage of ‘on’, ‘at’ and ‘in’ /usr/bin vs /usr/local/bin on Linux Frequency of the sound when blowing in a bottle Not enough free disk space when upgrading Postfix Connection Timed out How can I improve Ubuntu overall system performance? How to configure Ubuntu as an LDAP client? How to configure Ubuntu 12.04 as LDAP client with PAM modules? Robust method to extract "pure" part of a variable I am trying to perform a multi linear regression model:  $$y_i = β_0 + β_1x_{i1} + β_2x_{i2} +... + β_px_{ip} + ε_i$$  where $$x_{i1}, x_{i2}, ..., x_{ip}$$ are highly correlated with each other (VIFs can be as low as 5 and high as 30).  I am just wondering if there exists a procedure with the following properties:  1) extracts the pure part of each independent variable, where pure is defined as a part of the variable which is uncorrelated with other variables.  2) the pure part and the original variable should be similar to each other, whether similar is defined as moments such as mean and variance and correlation coefficient (ideally >0.90) Creating a unique alphanumeric 10-character string I'm looking to create a simple short-lived reservation system, and I'd like to generate confirmation numbers that are   unique random-looking alphanumeric short-ish, at least much shorter than 32 character-long strings returned by sha1   I'm only looking to have ~500 reservations, so I don't imagine high likelyhood of collissions.  One idea I had is generate an sha1 hash based on a date-time stamp and username, then truncating it to its first 10 characters.  Would something like that be reliably unique enough for the purposes of processing ~500 reservations? What has been proved about the big bang, and what has not? Ok so the universe is in constant expansion, that has been proven, right? And that means that it was smaller in the past.. But what's the smallest size we can be sure the universe has ever had?  I just want to know what's the oldest thing we are sure about. Problem. Use Tutte's 1-factor theorem to prove that every 3-regular graph with no bridges has a 1-factor.  I honestly have no idea how to approach this problem. I have drawn a couple of graphs to gain some insight to no avail. Any hints or tips will be greatly appreciated. What is the purpose of the Assemblies node in Web.Config? When creating a new ASP.NET AJAX Web Application in Visual Studio 2005, the default web.config contains the following section (inside the compilation node):  &lt;assemblies&gt;     &lt;add assembly="System.Web.Extensions, Version=1.0.61025.0, Culture=neutral, PublicKeyToken=31bf3856ad364e35"/&gt; &lt;/assemblies&gt;   Whether or not the assemblies node exists doesn't seem to affect the AJAX functionality.  Update panels, etc work fine regardless of whether or not the reference to System.Web.Extensions is included in Web.Config.  What's the purpose of this section, and can it be safely removed? How to redirect users to custom lostpassword page? Is a visa to Poland required for a Russian spouse of a Polish citizen? I am a holder of a Russian passport and my husband is a Polish citizen. We both live and work in the UK. I have received my family member of EEA national card, good for 5 years. We would like to travel to Poland together. On the website of the Polish embassy it clearly states that if we are traveling together, I do not require a visa, however in the embassy they were sure that I do need a visa. Could anyone please clarify?