I'm supposed to use a prime factorization somewhere, and that the fundamental theorem of arithmetic is to be applied as well.
Show that if $n$, $a$, $b$, and $c$ are positive integers with $\gcd(a, b) = 1$ and $ab = c^n$, then there are positive integers $d$, and $e$ such that $a = d^n$ and $b = e^n$.   I know that (by Bezout) $\gcd\left(a,b\right) = 1$ implies $ax + by = 1$ for some integers $x$ and $y$, and also that $\gcd\left(a^n,b^n\right) = 1$, but this does not help me.
London Heathrow airport - do I need a transit visa? I am traveling from the US to India via London Heathrow on November 6. I will be at LHR on 7th at 6:20 am and departing at 9:25 pm. I'm an Indian national and have a valid US visa.  Do I need a transit visa to stay airside?  I have been scared that British Airways have stopped a few people from boarding flights (no details just rumors).  Do I need a transit visa to stay landside? I want to do some sightseeing. What are the chances of getting past officials using transit without visa concession?  Real stories will help a lot to calm the butterflies in my stomach. Why KL divergence is non-negative? Why is KL divergence non-negative?  From the perspective of information theory, I have such an intuitive understanding:  Say there are two ensembles $A$ and $B$ which are composed of the same set of elements labeled by $x$. $p(x)$ and $q(x)$ are different probability distributions over ensemble $A$ and $B$ respectively.   From the perspective of information theory, $\log_{2}(P(x))$ is the least amount of bits that required for recording an element $x$ for ensemble $A$. So that the expectation $$\sum_{x \in  ensemble}-p(x)\ln(p(x))$$ can be interpreted as at least how many bits that we need for recording an element in $A$ on average.   Since this formula puts a lower bound on the bits that we need on average, so that for a different ensemble $B$ which brings about a different probability distribution $q(x)$, the bound that it gives for each element $x$ will surely not bit that is given by $p(x)$, which means taking the expectation, $$\sum_{x\in ensemble}-p(x)\ln(q(x))$$ this average length will surely be greater than the former one, which leads to $$\sum_{x\in ensemble }p(x)\frac{\ln(p(x))}{\ln(q(x))} &gt; 0$$  I don't put $\ge$ here since $p(x)$ and $q(x)$ are different.  This is my intuitive understanding, is there a purely mathematical way of proving KL divergence is non-negative? The problem can be stated as:  Given $p(x)$ and $q(x)$ are both positive over real line, and $\int_{-\infty}^{+\infty}p(x)dx = 1$, $\int_{-\infty}^{+\infty}q(x)dx = 1$.  Prove $$\int_{-\infty}^{+\infty}p(x)\ln\frac{p(x)}{q(x)}$$ is non-negative.  How can this be proved? Or can this be proved without extra conditions? A student prepares for an exam by studying a list of 12 problems. She can solve 9 of them. For the exam, the instructor selects 7 problems at random from the 12 on the list given to the students. What is the probability that the student can solve 5 of the 7 problems on the exam?  I am really confused by this problem. My first initial thought was to do:  (5 chose 7)(7 chose 0)/ (12 chose 7)  I am not 100% sure about this though. Function pointers/delegates in Java? For my Java game server I send the Action ID of the packet which basically tells the server what the packet is for. I want to map each Action ID (an integer) to a function. Is there a way of doing this without using a switch? Could anyone tell me what is up with so many archives, and how are they differ?  There's at least five that I can think of, .tar.tz, .tar.gz, .tar.bz2, .tar.lz, .tar.z  Which is best for what applications? It just seems silly to make so many variations of the same file but I'm sure there's a reason. I was trying to think but cannot figure it out. For instance, if the interaction is small, for instance limited to a windows, the observers in each universe will see that the other goes in reverse. But what if they can start a more meaningful interaction? (for instance, if one observer crosses the window and moves to the other universe, will anything change? why and how fast? (at least at the beginning his thinking will move in the same time arrow than in his universe, but the physical processes around him will still violate the second law? How to search for questions and answers between two users? I am searching for the questions posted by me (user:139150) that were answered by another user.    His ID is: 55159    There does not seem to be any way to search for posts where both the users are involved. Feeling a tingling sensation by touching stuff, Is this safe? I'm putting on my desktop computer on steel rack as usual. A while ago, when I touch the steel rack for seconds, practically I feel a tingling sensation, sort of electrical shock in my finders. After that I measured how voltage the steel rack has, the multi-tester meter indicated almost 50V by AC mode. Is this safe to touch or keep the PC on the position, or I should get away the PC somewhere electricity-safe? I'm not sure this site is suit for asking this kind of question, but I'm very worrying whether this is safe or not since this experience is my very first time. Thank you. I don't like having the middle mouse button paste, because I often end up with uncompilable code in Eclipse. How can I turn this off (in all programs)? I'm running Fedora. Typesetting Chemical Formulas Proving $A+A'B=A+B$ without truth tables How can I prove the Boolean algebraic rule $$A+A'B=A+B$$ without using a truth table?  With the truth table, it is easy to see that the two are equal, but how can I prove it using lesser Boolean identities? Difference between "git add -A" and "git add ." The command git add [--all|-A] appears to be identical to git add .. Is this correct? If not, how do they differ? Can someone explain the dollar sign in Javascript? The entire answer is moving when hovering the bounty button It's just a minor CSS issue. See the following gif: I just heard about the . I'd like to update my firmware, but I don't use Windows/OSX. How can I find out what version of Intel ME I have and how do I update it? I recently upgraded Windows 7 to Windows 10. The biggest issue I have now is that the Windows Search service no longer works, making the Start menu mostly unusable.  If I go to Indexing Options in the control panel I see that Indexing is not running and the list of paths to index is empty.  Going to advanced options, the current index path is empty, and the new path is empty too. This is true even if I try to use the "New Index" button.  Using the troubleshooter won't help since the service doesn't start. If I try to manually start the "Windows Search" service, I get the error  "Error 3: Unable to find specified path" (this is the rough translation, the original message is in my language).  What can I do to find out which path/file is missing ? I tried to check with Process Monitor, but the only file (beside system libraries) the SearchIndexer.exe try to access seem to be c:\ProgramData\Microsoft\Search\Data\Application (everytime the service is started the whole folder c:\ProgramData\Microsoft\Search is deleted and recreated) and C:\Windows\System32\config\systemprofile (which exists).  Edit: all available updates are installed, and issuing      SFC /scannow   Doesn't reveal any issue "Results found containing ___" too close to header Is there a word for happiness made possible by a tragic situation? How to combine multiple commands in terminal? I have the following commands.  cd import zcat urls1.sql.gz | mysql -u root -p urls cd /var/www/project1/ nano 1.php   As of now I'm executing it one by one.  Is there a way to combine those commands in one line? Frequency encoding is a widely used technique in Kaggle competitions, and many times proves to be a very reasonable way of dealing with categorical features with high cardinality. I really don't understand why it works.   Does it work in very specific cases where frequencies are correlated with the target or is it more general? What's the rationale behind it?