For a composite number $n, n=rs,$ where $n&gt;s≥r≥1.$  Show that $2^r-1$ divides $2^n-1$. Thank you.
Let $x &gt; 1$ and $a$, $b$ be positive integers. I know that $a$ divides $b$ implies that $x^a - 1$ divides $x^b - 1$. If $b = qa$, then  $$x^b - 1 = (x^a)^q - 1^q = (x^a - 1)((x^a)^{q-1} + \ldots + x^a + 1).$$  I'm interested in the converse of the statement. If $x^a - 1$ divides $x^b - 1$, does this imply that $a$ divides $b$?
How to change keyboard layout via command line cmd.exe on Windows XP/7? Is is possible to change the keyboard layout via command line on Windows XP/7? crisscross, dillydally, riffraff, etc Some English words only differ in their vowels: crisscross, dillydally, riffraff, etc. Is there a name for them? How did half of Darth Maul survive? The propagation of electric field which vectors are perpendicular to each other?  $\vec a = (1, -2, 3)$, $\vec b = (5, 4, 1)$, $\vec c = (1, 0, -5)$  Do i just take the dot product of 2 of them. If the dot product they are at $90^\circ$? But how do i know if there perpendicular? Why do people omit the definite article? I'm used to always hearing or seeing a  before certain nouns.  Yet on certain occasions the article is totally omitted, and it bothers me.  I'm wondering what the justification for omitting the article is, especially when the definite article would seem to be required, or if it is just bad English either on the part of the individual in question or more likely by tradition.  The most frequent place I hear this is at the doctor or dentist's office. In every one I have ever visited, the receptionist or assistant always says things like "Doctor will see you shortly" instead of "The doctor will see you shortly."  The wikipedia article linked above says this: "A definite article indicates that its noun is a particular one (or ones) identifiable to the listener".  In the case of a clinical setting, I know who the doctor is, especially if this is not my first visit.  The people in the office know the doctor even better than I, so it would seem to be inappropriate to omit the article. Even the indefinite article would be incorrect. It further says, under "Zero article":  "In languages having a definite article, the lack of an article specifically indicates that the noun is indefinite."  Since the noun is not indefinite, why are they omitting the article?  Another place I see the article omitted is with the word "bar".  Bartenders apparently don't say "I tend the bar" they say "I tend bar".  Unless a bartender works in multiple bars, it would seem like an article would be required.  Furthermore, "the bar" can mean any bar a person goes to, so even in a case like , it seems like the word "the" should be in the sentence.  The only thing I can think of in this case is maybe the speaker doesn't want to confuse the listener into thinking they work in the legal field.  Is this just a bad habit, a historical corruption or tradition, or is there really a valid reason (or perhaps more than one) for these omissions? How do you refer to a hyponym that is the same word as the hypernym? What word (or how do you phrase things) do you use when the ostensible word for the class is the same as the word for a subset of the class?  For example, in the United States, there are many brands of sugared, carbonated soft drink that one can buy: Coca-Cola (or Coke), Pepsi, RC Cola, Sprite, 7-Up, A&amp;W or Mug Root Beer, etc). The first two are kinds of 'coke' or 'cola' (but no one says 'cola').  , one asks for any type of such drink as 'coke'. How does one then ask for the particular kind of 'coke' that is 'Coke' (pronounced the same)? Does one reduplicate and say a 'Coke coke', does one repeat with emphasis (or without), does one use a different word like 'coca-cola' (co-cola), or what?  The (perceived) difficulty (surely those in the South are able to get the drink they want somehow) isn't limited to there. Where the drink is called 'soda', if you ask for a 'coke', you'll still sometimes have the choice among Coca-Cola, Pepsi, etc. "Can I have a Coke?" the response might be "Is Pepsi OK?". "Sure, Pepsi, Coke, whatever." You don't use 'cola', you just ask for the particular kind and hope the listener will help differentiate.   Given these multiple strategies for disambiguating, what do you do say to get a 'Coke' in the southern US? are there any other examples of a hypernym being the same as the hyponym, and so creating a disambiguation problem? How do I split a string so I can access item x? The locus of the intersection point of two perpendicular tangents to a given ellipse What does the caret (‘^’) mean in C++/CLI? Proof that the irrational numbers are uncountable Blog showing up as mobile, won't exit Stack Exchange blog is showing me the mobile site and won't let me exit on Chrome 21. Could someone take a look? Thanks. Edit I noticed it was with the special query string, that I can't exit it. How does friction stop a car? I am looking for a preferably introductory text on Yang-Mills theory with mathematical basis and language. (Differential forms) I have a background in QCD and QED, however, I want to go more in details in geometrical interpretations of Yang-Mills theory. I am sorry if this question has asked before. Thanks in advance. How do I prove that $S^n$ is homeomorphic to $S^m \Rightarrow m=n$? This is what I have so far:  Assume $S^n$ is homeomorphic to $S^m$. Also, assume $m≠n$. So, let $m&gt;n$.  From here I am not sure what is implied. Of course in this problem $S^k$ is defined as:  $S^k=\lbrace (x_0,x_1,⋯,x_{k+1}):x_0^2+x_1^2+⋯+x_{k+1}^2=1 \rbrace$ with subspace topology. I was hoping I would find some people in the industry that could help me decide what's the best price to charge to design (in this case) business cards.  What are standard prices for professional designers?  From a firm point of view? From a freelancer?  I don't want to overcharge clients but I don't want to ruin the industry by underselling my services. Can every game of Klondike-Solitaire be solved? I play a lot of Solitaire games on my Android phone and love to keep an eye out for the statistics.  Given that the Solitaire version lets you restart the game endlessly, I usually play until I solve it. But I never managed to solve more than 80% of the games played (1000+).  So now I wonder, is every Solitaire game solvable? At college we use ArcGIS Desktop 10, but that does not function on my Mac so I have QGIS there instead.  I want to do my exercises with QGIS, but in ArcGIS Desktop we use .lyr files with ArcMap and .lyrx files with ArcGIS Pro, and they are not supported by QGIS.  Does anyone know, how I can connect the *.lyr files to my *.shp files to use them on my Mac with QGIS? How do we compare the size of numbers that are around the size of Graham's number or larger? When numbers get as large as Graham's number, or somewhere around the point where we can't write them as numerical values, how do we compare them?  For example:  $$G&gt;S^{S^{S^{\dots}}}$$  Where $G$ is Graham's number and $S^{S^{S^{\dots}}}$ is $S$ raised to itself $S$ times and $S$ is Skewes number.  It appears obvious (I think) that Graham's number is indeed larger, but how does one go about proving that if both numbers are "so large" that they become hard to compare?  More generally, how do we compare numbers of this general size?  As a much harder problem than the above, imagine a function $G(x,y)$ where $G(64,3)=$ Graham's number.  The function $G(x,y)$ is as follows:  $$G(x,y)=y\uparrow^{(G(x-1,y))}y$$  Where $G(0,y)$ is given.  I ask to compare $G(60,S)$ and $G(64,3)$ Trying to install CPAN