What is the value of $187^2 \mod 100$?
I was wondering if there is a simpler way to find the last to digits of a power such as $3^{45}$. I reduced it modulo 100 to get the answer, which is 43. But I was curious if there was a simpler, or more eloquent way to get this result.  Here's my method:  Using the fact that $3^4=81\equiv -19 \pmod{100}$ and $19^5=2,476,099\equiv -1 \pmod{100}$  $$ 3^{45}\equiv 3(3^4)^{11}\equiv 3(-19)^{11}\equiv 3(-19)(19^5)^2\equiv 3(-19)(-1)^2 \equiv -57 \equiv 43 \pmod {100} $$  I just felt like this was sort of a messy way to find out that the last two digits are 43.
find an approximate solution, up to the order of epsilon The question is to find an approximate solution, up to the order of epsilon of following problem.  $$y'' + y+\epsilon y^3 = 0$$ $$y(0) = a$$ $$y'(0) = 0$$  I tried to solve the given problem using perturbation theory.  $$y(t) = y_0(t) + \epsilon y_1(t) + \epsilon^2 y_2(t) + \cdots$$ $$t = w(\epsilon)t = (1 + \epsilon w_1 + \epsilon^2 w_2 + \cdots )t$$  However, i failed to find an appropriate approximate solution... help me!! How to convert java.util.Date to java.sql.Date? I am trying to use a  as input and then creating a query with it - so I need a .    I was surprised to find that it couldn't do the conversion implicitly or explicitly - but I don't even know how I would do this, as the Java API is still fairly new to me. Deleting an element from an array in PHP Is there an easy way to delete an element from an array using PHP, such that foreach ($array) no longer includes that element?  I thought that setting it to null would do it, but apparently it does not work. Detecting and acting on keyboard direction keys in Java Is there a Windows standard way to do things such as "start fan", "decrease speed" or the like, from C/C++?  I have a suspicion it might be ACPI, but I am a frail mortal and cannot read that kind of documentation.  Edit: e.g. Windows 7 lets you select in your power plan options such as "passive cooling" (only when things get hot?) vs. "active cooling" (keep the CPU proactively cool?). It seems the OS does have a way to control the fan generically. Please give me feedback on my answer to this question. Question: For all $ n\geq1:\binom{2n}{0}+\binom{2n}{2}+\binom{2n}{4}+\cdots+\binom{2n}{2k}+\cdots+\binom{2n}{2n} $   is equal to $ \binom{2n}{1}+\binom{2n}{3}+\binom{2n}{5}+\cdots+\binom{2n}{2k+1}+\cdots+\binom{2n}{2n-1} $. Answer: False, since;  Let $a=1,  b=-1$  and $n=1$  ,$k=0$   , using Binomial expression.  $$(a+b)^n=  \sum   \binom{n}{k}  a^{k}b^{n-k}$$  Then $(1+-1)^{2(1)}=\binom{2}{0}1^{0}(-1)^{2-0}+\binom{2}{1}1^{1}(-1)^{2-1}+\binom{2}{2}1^{2}(-1)^{2-2}=\binom{2}{0}-\binom{2}{1}+\binom{2}{2}$  Thus :For all $n\geq1:  \binom{2n}{0}+\binom{2n}{2}+\binom{2n}{4}+\cdots+\binom{2n}{2k}+\cdots+\binom{2n}{2n}   \ne \binom{2n}{1}+\binom{2n}{3}+\binom{2n}{5}+\cdots+\binom{2n}{2k+1}+\cdots+\binom{2n}{2n-1}$  . I'm a bit confused about the concept of ground, and perhaps voltage as well, particularly when trying to analyze a circuit. When I learned about Ohm's law in grade school, I learned how to apply the law to calculate current, voltage, and resistance of simple circuits.   For instance, if we were given the following circuit:    We would be could be asked to calculate the current passing through the circuit. At the time, I'd simply compute (based on the rules given) 1.5V/1Ohms=1.5A.  Later on, however, I learned that the reason the voltage of the resistor would be 1.5V is because voltage is really the difference in potential between two points, and that the difference of the voltage across the battery would be the same as that of the resistor (correct me if I'm mistaken), or 1.5V. I got confused, however, after the introduction of the concept of ground.   The first time I tried to do the current calculation for a circuit similar to the previous circuit on a simulator, the program complained about not having a ground and "floating voltage sources". After a bit of searching, I learned that circuits need ground as a reference point or for safety reasons. It was mentioned in one explanation that one can pick any node for ground, although it's customary to design circuits so there is a "easy place" to pick ground.   Thus for this circuit    I picked ground at the bottom, but would it be okay to pick ground between the 7 ohm and 2 ohm resistor - or any other place? And what would be the difference when analyzing the circuit?  I've read that there are 3 typical ground symbols with different meanings - chassis ground, earth ground, and signal ground. A lot of circuits I've seen used in exercises either use earth ground or signal ground. What purpose is there in using earth ground? What is the signal ground connected to?  Another question: since the ground is at unknown potential, wouldn't there be current flowing to or from ground to the circuit? From what I've read we treat the ground as 0V, but wouldn't there be some sort of effect because of a difference in potential of the circuit and ground? Would the effect be different depending on what ground was used?  Finally: In nodal analysis, one customarily picks a ground at the negative terminal of the battery. However, when there are multiple voltage sources, some of them are "floating". What meaning does the voltage of a floating voltage source have? After installing the latest Ubuntu I find that the icons in the applications folder (viz. /usr/share/applications), when opened in the Files application (Nautilus), come up as a blank page with a gear in the middle. If I try to open them I get a read only text box. How to get color of pixel in coordinates 123, 456 in screen? My AppleScript app needs to get color of pixel in coordinates 123, 456 in screen. Default currency symbol I'm looking for a way to get the Default currency symbol (in a none-multi-currency org) of the salesforce org locale (to have it displayed on a visualforce page in a field that can't be &lt;apex:inputField&gt; or &lt;apex:outputField&gt;) preferably query-able from apex (via the database, a describe or whatever) and then again the actual symbol, meaning € for Euro, $ for dollar, and so on...  There must surely be some sort of mapping hidden in the code somewhere salesforce does this via the input/outputfields if bound to an sobject of type currency  Can anyone point me in the right direction? Infinitely differentiable function with given zero set? Are all stormtroopers as poor soldiers as the ones in the movies? In the Star Wars films we consistently witness just how poor soldiers stormtroopers are. They show very sub-par marksmanship, crappy house searching tactics, and are easily fooled and defeated by Ewoks and Rebels.   That being said, I remember reading a character guide that defined stormtroopers as the very best of the best (best marksman, above average intelligence, etc.). So, in the realm of the Expanded Universe, does the mold that the movies set continue to hold true or are they just the exceptions to the rule? Why did the Jedi want to bring balance to the Force? During the Prequel trilogy, the Jedi wanted to bring balance to the Force, in accordance with a prophecy.  There seemed to be a general agreement that this was a 'good thing', but a disagreement with the assertion that Anakin was the one to do it.  Why did the Jedi Council believe it was good to bring balance to the Force?  The Republic was mighty (in theory), the Council was established as THE source of Force users, the Sith were thought extinct, and the number of Dark Jedi was apparently very, very low.  It seemed, for all intents and purposes, as if the Force was strongly imbalanced in favor of the Light side...so why did the Council want to 'balance' it? iBooks library like effect How to get my 4yo daughter's fixation on my belly button under control? I don't know where to begin with this question. It's awkward and to be honest, it makes me feel rather uncomfortable. Simply put, I'm led to believe that my four year old daughter is obsessed with my belly button. It all started one day during the summer when she was three. I decided to take my family to the pool, and I guess I didn't see my daughter follow me into the changing room. Mind you, she's still at the age where it's appropriate to see her parents naked, so I wasn't surprised when, after I finished changing into my bikini, I turned around to see her staring at me.  I followed her gaze and saw her looking directly at my belly button. I asked her what she was looking at and she screamed "mommy, I can see your belly button!" which made me giggle. I picked her up and kissed her forehead, telling her that everyone has a belly button.  I think I might have accidentally given her a green light, because over the next year she constantly asks me if she can stick her finger in my belly button, and if I don't let her, she begins to scream.  For her fourth birthday, she requested that I wear nothing but a sports bra and underwear (in toddler speak. It sounded like "mommy, show your belly button for my birthday!" I could have simply worn a crop top but i didn't want to risk her throwing a temper tantrum)  My husband obviously doesn't mind, because he gets to look at my belly button all the time but it really upsets me that my daughter's finger is constantly in my belly button.  Tomorrow is her fifth birthday and I want to discuss with her that it's not appropriate to always be poking my belly button. Any ideas on how i can talk to her without her throwing a screaming fit? I am writing a small matrix library in C++ for matrix operations. However my compiler complains, where before it did not. This code was left on a shelf for 6 months and in between I upgraded my computer from debian etch to lenny (g++ (Debian 4.3.2-1.1) 4.3.2 ) however I have the same problem on a Ubuntu system with the same g++.  Here is the relevant part of my matrix class:  namespace Math {     class Matrix     {     public:          [...]          friend std::ostream&amp; operator&lt;&lt; (std::ostream&amp; stream, const Matrix&amp; matrix);     } }   And the "implementation":  using namespace Math;  std::ostream&amp; Matrix::operator &lt;&lt;(std::ostream&amp; stream, const Matrix&amp; matrix) {      [...]  }   This is the error given by the compiler:     matrix.cpp:459: error: 'std::ostream&amp;   Math::Matrix::operator&lt;&lt;(std::ostream&amp;,   const Math::Matrix&amp;)' must take   exactly one argument   I'm a bit confused by this error, but then again my C++ has gotten a bit rusty after doing lots of Java those 6 months. :-) I've been try to download Xcode for work, but for some reason the Mac App Store is giving me a bit of trouble. Activity monitor shows some network usage, but in Launchpad it seems the download isn't moving.   When I go to the purchases panel of the Mac App Store, I see this.     No status, no pause button, in fact - the button has no text on it! I've tried restarting the App Store, restarting my computer, even removing the com.apple.appstore folder in /var/folders but alas, no luck.  Can anyone diagnose the issue? When I removed com.apple.appstore, Xcode still showed up on Launchpad, and clicking on it did nothing - it was frozen in time. So can I kill off this partial download and start new? Is it true that if $f(x)$ has a linear factor over $\mathbb{F}_p$ for every prime $p$, then $f(x)$ is reducible over $\mathbb{Q}$? We know that $f(x)=x^4+1$ is a .  My question is:     Is it true that if $f(x)$ has a linear factor over $\mathbb{F}_p$ for every prime $p$, then $f(x)$ has a linear factor over $\mathbb{Q}$?   Edit: Thanks for @Jyrki Lahtonen's answer, I want to do some modifications:     Is it true that if $f(x)$ has a linear factor over $\mathbb{F}_p$ for every prime $p$, then $f(x)$ is reducible over $\mathbb{Q}$?   Thanks in advance! How to catch and print the full exception traceback without halting/exiting the program? How can a distribution have infinite mean and variance?