Let a, b, c ∈ N such that (a, b) = 1 and a | bc. Prove that a | c.  I'm a little confused about if I'm doing this proof right.  I know that $\exists  p, q \in \mathbb{Z}$. Such that $pa = bc$ and $qa=c$. Re-arranging the first equation.  $$a = \frac{bc}{p} $$  Substituting this into the second equation.  $$ qa = c$$ $$ q(\frac{bc}{p}) = c$$ $$ \frac{qb}{p} c =c $$ $$ \frac{qb}{p} c =c $$  Thus this equation divides c. Therefore a | c. Does this proof make sense.  Thank you for any guidance.
Well I thought this is obvious. since $\gcd(a,b)=1$,  then we have that $a\not\mid b$ AND $a\mid bc$. This implies that $a$ divides $c$. But apparently this is wrong. Help explain why this way is wrong please.   The question tells you give me two relatively prime numbers $a$ and $b$ such that $a$ divides the product $bc$, prove that $a$ divides $c$. how is this not obvious? Explain to me how my "proof" is not correct.
Display number with leading zeros Given: a = 1 b = 10 c = 100  How do I display a leading zero for all numbers with less than two digits? This is the output I'm expecting: 01 10 100 How to specify the SMTP server in PHPMailer? Which "mean" to use and when? So we have arithmetic mean (AM), geometric mean (GM) and harmonic mean (HM). Their mathematical formulation is also well known along with their associated stereotypical examples (e.g., Harmonic mean and it's application to 'speed' related problems).  However, a question that has always intrigued me is "how do I decide which mean is the most appropriate to use in a given context?" There must be at least some rule of thumb to help understand the applicability and yet the most common answer I've come across is: "It depends" (but on what?).  This may seem to be a rather trivial question but even high-school texts failed to explain this -- they only provide mathematical definitions!  I prefer an English explanation over a mathematical one -- simple test would be "would your mom/child understand it?" How to import Apple Mail archive files in Outlook 2013? I need to import Apple Mail archived files in Outlook 2013 on Windows, but every time I open the import wizard and reach the file selection step, I cannot select the archived file. It doesn't even show in the dialog box.  I don't know what the issue is. Is it the file size issue as the archive file is large (about 7 GB) or there is any other fault? Show $100+t$ is reducible in $\mathbb{Z}[[t]]$ $\mathbb{Z}[[t]]$ is the ring of formal power series with integer coefficients, and the problem asks to show that $100+t$ is reducible. This means I need to find two power series, $\sum_{i=0}^\infty a_i t^i$ and $\sum_{j=0}^\infty b_j t^j$ with $a_i,b_j\in\mathbb{Z}$, that when multiplied give $100+t$. Since the integers are an integral domain, I know that neither of these power series may be finite, but I'm having a really hard time figuring out how to come up with two series that will telescope properly. Any suggestions? Select multiple faces at once I have a .STL file and all of the faces have been converted to triangles so there are hundreds of long thin triangles.  Is there a effective way to select all of the faces in a small area? Something like a circle and when you click every face under that circle is selected. How to return to the GUI? I am very new to Linux and Ubuntu. I use Ubuntu 11.10.  I was trying to exit the X server (in order to install some Nvidia drivers) and I typed in the terminal:  sudo service lightdm stop   How can I return to the Ubuntu interface? How can I let a footnote continue on the next page? I have a rather long footnote which breaks a listing in two parts, unnecessarily.  Since the listing is more important then the footnote I'd rather lake to have the footnote split.   Can I customize (one or all) footnotes to have a maximal length before they are continued on the next page? Or can I manually split a footnote in two to get the effect? How can I "grep" patterns across multiple lines? It seems I am misusing grep/egrep.   I was trying to search for strings in multiple line and could not find a match while I know that what I'm looking for should match. Originally I thought that my regexes were wrong but I eventually read that these tools operate per line (also my regexes were so trivial it could not be the issue).  So which tool would one use to search patterns across multiple lines? Intuition behind the Definition of Conditional Probability (for 2 Events) Are parallel calls to send/recv on the same socket valid? I am travelling to Budapest and therefore have a Schengen Visa. I hold an Indian passport and a valid US visa. I'm travelling from Chicago-Heathrow-Budapest and returning by Budapest-Dublin-Chicago.  Would I need a transit visa for UK/Ireland?  UPDATE  Hoping this will help someone someday.   I was not allowed to board my return flight (BUD-DUB-ORD) because I do not hold an overnight Irish Transit Visa. They said I could have transited happily if the connecting flight was in the same day, but because I would have to originally land at 10 PM and catch the connecting flight next morning at 10 AM, I needed a transit visa. Well, I thought it was BS, but couldn't help comply with the rules. I booked another last minute ticket and flew via Warsaw via LOT. Curved mesh topology 2 I am trying to model a violin belly and I am intrigued, how is it possible to make faces with soft / rounded edges?    The edges I do are totally straight (see the blue amoeba).  Also, do you have some idea on how to avoid the edge loops getting too "compressed" in the middle of the model (see the red amoeba) ?    I know the Subdivision surface modifier, but I want curved edges after applying the modifier... like in this picture :  Any help will be very appreciated. Thank you in advance!  Marcos  2nd part  In fact, I've tryied to model using nurbs and bezier curves, but I wasn't able to find any tool similar to loft that works with curves, then I was forced to convert it in meshes, the results were not terrific and I've started to move the vertices manually, a real pain... Unlike a guitar, each edge loop is on a diferente elevation on z axis and I have to keep the original dimensions, So I have to be extremely careful when using the subdivision suface modifier. I've already modeled this think on three different CAD softwares, but the meshes are far from good... I want to improve the edge loops, but it will be a long path... Thank you again. Show that $A \setminus ( B \setminus C ) \equiv ( A \setminus B) \cup ( A \cap C )$ Show that  $A \setminus ( B \setminus C ) \equiv ( A \setminus B) \cup ( A \cap C )$  This is an exercise I was trying to do ( not homework ) and I got stuck as follows:  Working from $A \setminus ( B \setminus C )$:  $x \in A \wedge x \notin ( B \setminus C )$ $x \in A \wedge ((x \in B \wedge x \in C) \vee (x \notin B \wedge x \in C) \vee (x \notin B \wedge x \notin C) )$  At this point I'm not sure how to proceed.  Working from the other side $ ( A \setminus B) \cup ( A \cap C ) $:  $ x \in ( A \setminus B) \cup ( A \cap C ) $  $ x \in ( A \setminus B) \vee x \in ( A \cap C ) $  $ ( x \in A \wedge x \notin B ) \vee ( x \in A \wedge x \notin C ) $  $ x \in A \wedge (x \notin B \vee x \notin C ) $  $ x \in A \wedge x \notin ( B \cap C ) $  $ A \setminus ( B \cap C ) $  I could also use some formating tips like how display logical not and how to get the lines to display properly without blank lines inbetween. :-) How much money do you need to have on hand when entering USA? My friend, a Chilean citizen, is entering the US via waiver program to spend 3 weeks in USA in a couple of weeks. She already got her ESTA approval, the plane ticket, all hotels are already booked (but paid by me, not her).  How much money would she need to bring with her when entering US? Do they still check this? She would of course have credit cards with her. Proving a combinatorial identity without double counting Consider the following problem:  We have a collection of m white and n black balls. We pick k balls (k ≤ m+n) without replacement. What is the expected number of white balls in the sample?  Counting method 1: The ratio of white balls in the collection is m/(m+n). If we repeatedly sample the collection, we expect that on average, our sample has the same ratio. Therefore, the expected number of white balls in a sample of size k is km/(m+n). As a side note, I think the above argument somehow depends on the unbiasedness of  the mean estimator. Am I right?  Counting method 2: We directly compute the expected number of white balls in the sample: $$ \frac{\sum_{i=0}^{k} i\binom{m}{i}\binom{n}{k-i}}{\binom{m+n}{k}} \enspace.$$  So, we reach at the identity: $$ \sum_{i=0}^{k} i\binom{m}{i}\binom{n}{k-i} = \frac{km}{m+n}\binom{m+n}{k} \enspace.$$  How to prove the above result without double counting? It seems to be provable using  as well as $i\binom{m}{i} = m\binom{m-1}{i-1}$. But is there a simpler approach (which does not depend on extra identities such as Vandermonde's)? What is the motivation behind axiomatically forcing the underpinning group of a ring to be abelian? Noncommutative rings are vastly more complex than commutative ones, so I am assuming that allowing the additive operation to be noncommuting would just make matters worse. Is there something deeper here, or is this a restriction for the sake of convenience and simplicity? My multiple regression analysis model has a statistically significant F value however all beta values are statistically non-significant.   All the regression assumptions are met. No multicollinearity was found. Correlations among all predictors are all less than 0.60.   What else might be the cause of the insignificant predictors? show that if $a | c$ and $b | c$, then $ab | c$ when $a$ is coprime to $b$. Given two numbers $a$ and $b$, where $a$ is co-prime to $b$,     Show that for any number $c$, if $a|c$ and $b | c$ then $ab| c$. Is the reverse also true? In other words, if $ab |c$ and $a$ is co-prime to $b$, then do we have $a | c$ as well as $b|c$? I want to create posters for my poster presentation on a conference. What tools or LaTeX classes are available for preparing posters ?