The highest common factor and lowest common multiple of two numbers $A$ and $B$ are $12$ and $168$ respectively. Find the possible values of $A$ and $B$ with the exception of $12$ and $168$.  I know this question involves the concept of  $ A \times B =  \text{lcm}\,(A,B)\times\text{hcf}\,(A,B)$   But I'm not sure how to apply it. Can someone guide me? Thanks!
The highest common factor and the lowest common multiple of two numbers $A$ and $B$ are $12$ and $168$ respectively. Find the possible values of $A$ and $B$ with the exception of $12$ and $168$.   Help me with this equation please. I kind of get it but I don't understand the concept behind it... I do not want to memorize.
Am wondering if somewhere there is a list of industry "minimums" for browser support. My company release two "portals" over the past 6 months and did very little to support IE8 users, who at this time, according to my G Analytics makes up about 14% of all IE visitors. Amazingly enough IE7 is 25%...  Figured support for IE7 was out the window already, but figured 14% was significant enough to warrant some support.  Any arguments to support my side of wanting to support IE8?  Any arguments to NOT support IE8? Proving a function is Lipschitz continuous Show that the following function is Lipschitz continuous and find a Lipschitz constant  $$y\mapsto f(x,y)\\ f(x,y)=\frac{y}{x}\ln(\frac{y}{x})\text{  , } |x-1|\leq\frac{1}{2}\text{  , } |y-e|\leq\frac{e}{2} $$  I have no clue on where to begin to prove this.   My first question is how I should interpret this function, is $x$ a sort of constant? iOS 10 remember to "not trust this computer" Concatenating numbers and adding in leading zeros? I have three text fields populated as follows: 1, 9, 1 that I need to concatenate into a field that looks like 01.09.01.  Getting to 1.9.1 is easy but can anyone help out with how to formulate it so that each value has two places (i.e. add in those zeroes).  I used to do this in Excel but can't figure out how to work in ArcMap 10. Ubuntu 18.04 can't resume after hibernate Hibernate worked correctly when I had been using Ubuntu 17.04. After upgrade to 17.10 I couldn't resume because of black screen after resume ().   After upgrade to 18.04 if I try to start my computer after hibernation it boots as if there was no hibernation done. Name of short sci-fi story involving a musician who is punished for listening to Bach Partition of circumference into $3k$ arcs The following problem is from 1982 Russian Mathematical Olympiad. If you go to this , and scroll down to the section Russian Math Olympiad, then this is Problem 333 in that text-file.     Let $k$ be a positive integer. There are $3k$ points marked on the circumference. They divide the circumference onto $3k$ arcs. Some $k$ of arcs have length 1, other $k$ of them have length 2, the rest $k$ of them have length 3. Prove that some two of the marked points are the ends of one diameter.   I haven't given too much thought into the problem yet (I probably wouldn't get anywhere), but I thought I would share this awesome problem.   Added. Trivial Observation: The length of the circumference is $k+2k+3k=6k$. My application got crashed because of not initializing the view How do you create grass? Change paper size in mid-document My document should be letterpaper and my attachments should be a4paper. I had high hopes for the \newgeometry command from the geometry package, but the documentation says      \newgeometry is almost similar to   \geometry except that \newgeometry   disables all the options specified in   the preamble and skips the   papersize-related options: landscape,   portrait and paper size options (such   as papersize, paper=a4paper and so   forth).   Any help is appreciated! One of the topics that seems to come up regularly on mailing lists and online discussions is the merits (or lack thereof) of doing a Computer Science Degree. An argument that seems to come up time and again for the negative party is that they have been coding for some number of years and they have never used recursion.  So the question is:   What is recursion? When would I use recursion? Why don't people use recursion? How do I unit test (using xUnit) a class that has internal private methods, fields or nested classes? Or a function that is made private by having  (static in C/C++) or is in a private () namespace?  It seems bad to change the access modifier for a method or function just to be able to run a test. Let's consider a polynomial ring of single variable. One can define them informally by saying $P(X)=\sum_{i=1}^n a_n X^n$ while $X$ is an indeterminate variable. However, since mathematics is based on first-order logic, one can only talk about something that actually exists (as a set). So as its name says, indeterminate variable is not a sentence in $ZFC$. Nevertheless, we can formally define a polynomial ring $R[X]$ as a subset of $R^\omega$ whose support is finite. ($R$ is a commutative ring with unity) Just like a single variable, I want to know what would be the formal definition of a polynomial ring of several variables. What would be a formal definition? === I have figured out a candidate. That is,  Candidate for a &quot;Definition of a polynomial ring $R[X_1,...,X_n]$&quot;. ($R$ is a commutative ring with unity) Let $R[X_1,...,X_n]$ be the set of all functions $f:\omega^n \rightarrow R$ whose support is finite. Define $(f+g)(a,b,c)=f(a,b,c)+g(a,b,c)$ and $(f•g)(a'',b'',c'')=\sum_{a+a'=a''} f(a,b,c)•g(a',b',c')$ (Note that in this way, $+,•$ are well defined.) Then, it can be checked that $(R[X_1,...,X_n],+,•)$ is a ring.  Is this a formal definition of a polynomial ring? Possible Duplicate:          Sounds like a silly question, but bear with me..  If you go into Peet's Coffee and access the wifi, you're taken to a web page that requires a 4-digit code you get with a purchase.   If you do the same at Starbucks you're taken to a page and asked to OK the terms of service.   I want to build something similar for a local independent coffee shop, something that requires the user to write a little about themselves before they can access the Internet.  I'm not sure which way to go -- looking for wifi routers that run an internal web server, or finding one that lets you direct users to an external website for authentication, somehow redirecting back to the access point once successful.   Has anyone done anything like this and can share their experience? How to serve static files in Flask Curve Bevel function, scale? Is there a way to scale up the bevel object on one end of a curve, a little bit over each point?  I notice the vertex "weight" slider will tweak how much influence each vertex has on the others, is there something like this that can scale the bevel object up a little bit?  Preferably with the option to constrain X,Y,Z independently Run Script on Wakeup? Switch off Windows Update for a particular program or feature I'm getting really tired of unchecking the same Windows Update candidates so save hundreds of MB's of dowbloads, especially for programs I never use, that are in fact not even installed, and those I don't wish to update.  Is there any way to permanently, or until another explicit change, to turn off certain updates?  There is an option called 'Hide update', but that unsettles me a bit. When to use IMG vs. CSS background-image? In what situations is it more appropriate to use an HTML IMG tag to display an image, as opposed to a CSS background-image, and vice-versa?  Factors may include accessibility, browser support, dynamic content, or any kind of technical limits or usability principles. $f(x)=x/1$ is not injective? In chapter 3 of Atiyah and Macdonalds Intro to Comm Algebra there is the statement  "We also have a ring homomorphism $f:A\rightarrow S^{-1}A$ defined by $f(x)=x/1$. This is not in general injective." (Note: $A$ denotes an integral domain and $S$ a multiplicatively closed subset of $A$.)  How could this possibly not be injective? If $f(x)=f(y)$ then $x/1=y/1$ so $1x=1y$ hence $x=y$. What am I missing here?