Let $a, b, d \in\mathbb{Z}$. Suppose that $d \mid ab$. Prove that there exists $a_1, b_1 \in \mathbb{Z}$ such that $d = a_1 b_1$ and $a_1 \mid a$ and $b_1 \mid b$.
I read this recently on the web and can't manage to understand it. Not homework -- I haven't done math homework for years.  If $d\mid ab$ then $d=d_1 d_2$, that $d_1|a$, that $d_2|b$ and, further, if $(a,b)=1$ then $(d_1,d_2)=1$.  I know it's the same question... if $a = a_1 a_2 ... a_n$, why can't $d$ divide some $a_i$ and only $a_i$?
Sum of odd Fibonacci Numbers Trying to prove that the sum of odd-index consecutive Fibonacci numbers is the next even-index Fibonacci number. I have a gap in my proof that I cannot figure out. I know that induction would be easier and I have already done it that way, I am looking to use alpha and beta. $$\sum_{i=1}^n F_{2i-1} = F_{2n}$$  $$\begin{align*} \sum_{i=1}^n F_{2i-1}&amp;=\sum_{i=1}^1  \frac{1}{\sqrt{5}}(\alpha^{2i-1}-\beta^{2i-1})\\ &amp;=\frac{1}{\sqrt{5}}\left(\sum_{i=1}^1\alpha^{2i-1}-\sum_{i=1}^1\beta^{2i-1}\right)\\ &amp;\;\;\vdots\\ &amp;=\frac{1}{\sqrt{5}}[(\alpha^n-\beta^n)(\alpha^n+\beta^n)]\\ &amp;=\frac{1}{\sqrt{5}}(\alpha^{2n}-\beta^{2n})\\ &amp;=F_{2n} \end{align*}$$ How can I assign the output of a command to a shell variable? I want to assign the result of an expression (i.e., the output from a command) to a variable and then manipulate it – for example, concatenate it with a string, then echo it.  Here's what I've got: #!/bin/bash cd ~/Desktop; thefile= ls -t -U | grep -m 1 &quot;Screen Shot&quot;; echo &quot;Most recent screenshot is: &quot;$thefile;  But that outputs: Screen Shot 2011-07-03 at 1.55.43 PM.png Most recent screenshot is:   So, it looks like that isn't getting assigned to $thefile, and is being printed as it's executed. What am I missing? Time Manager (QGIS) date format I've been through associated answers but cannot quite seem to find something which works!  I'm having trouble getting TimeManager to recognise a simple data field (this is in QGIS2 - but I've had this error in earlier versions). I have these fields formatted in what appears to be one of the acceptable formats:    ... here is a screenshot of the field properties:    The format appears to be recognised in QGIS as I've worked out days difference between two dates (the "Time on Market" field shown in the image).  However ... I always end up with:   The Twitter.shp file works fine as a test, so I don't think this is an issue with the install / phyton conflict.  Any thoughts welcome! css vertical centering how could i vertically center a &lt;div&gt; within a &lt;div&gt; ?  my code so far:  &lt;div style="height:322px;overflow:auto;"&gt;     &lt;div style="border: Solid 1px #999999;padding:5px;"&gt;     &lt;/div&gt; &lt;/div&gt;   i have tried "top:50%;" and "vertical-align:middle;" without success  EDIT: okay so it's been discussed a lot. and i've maybe started another mini flame war. but for argument sake, how would i do it with a table then? i've used css for everything else so far so it's not like i'm not trying to employ "good practices".  EDIT: the inner div does not have a fixed height Illustrator: Is there an efficient and accurate way to join two circles by their tangents (and keep it editable)? It's easy to join two circles if they are the same size and horizontal/vertical, into a pill shape:    If the circles are at an add angle then it's still possible but you can no longer rely on smart guides.     You can add lines roughly where the tangents should be and then use the scissors tool on the circles, cut out the inner area and then join the two crescents into one shape.    Is there a better way to do this? Is there a way to do this that is still editable afterwards too?  There are some other questions that cover the same area as this but don't cover editability or joining circles that are not in line with each other (where smart guides can't help). Is it possible to print to the Report window in the Info view? I installed Ubuntu on a Dell XPS 13 laptop that had Windows 10 preinstalled on it. I installed it from a USB flash drive on a partition. When I boot my computer I can only boot into Windows 10 and Ubuntu is nowhere to be seen.  If I boot into my USB then I can see that Ubuntu is installed, but I can't get to it from the BIOS boot menu. How can I re-enable backspace navigation in Chrome? Can anybody explain the UNABLE_TO_LOCK_ROW error? Error Message:     Developer script exception from Nicomatic : quotepotential :   quotepotential: execution of AfterUpdate caused by:   System.DmlException: Update failed. First exception on row 0 with id   a0Q90000005PoxtEAC; first error: UNABLE_TO_LOCK_ROW, unable to obtain   exclusive access to this record: [] Trigger.quotepotential: line 14,   column 1      Apex script unhandled trigger exception by user/organization:   00590000001A6jh/00D90000000bovq      quotepotential: execution of AfterUpdate      caused by: System.DmlException: Update failed. First exception on row   0 with id a0Q90000005PoxtEAC; first error: UNABLE_TO_LOCK_ROW, unable   to obtain exclusive access to this record: []      Trigger.quotepotential: line 14, column 1   these mails are getting to my mail..........  trigger quotepotential on Quote_Line_Item__c (after insert,after update) {     Set&lt;Id&gt; quoteIds = new Set&lt;Id&gt;();     List&lt;Quote__c&gt; quotes = new List&lt;Quote__c&gt;();     for (Quote_Line_Item__c record: Trigger.new) {         if (record.Quote1__c != null){             quoteIds.add(record.Quote1__c);         }     }     for(AggregateResult ar:[                 SELECT                      Quote1__c,                      SUM(Max_Batch__c)sumMax                  FROM Quote_Line_Item__c                  WHERE Quote1__c=:quoteIds                  GROUP BY Quote1__c                 ]) {          quotes.add(new Quote__c(              Id=(Id) ar.get('Quote1__c'),              Potential__c = (Decimal) ar.get('sumMax')              ));     }     if (quotes.isempty() == false) {         update quotes;     } } hostname returns another device's name I realized that the hostname had changed while I was doing some work using terminal in OS X Yosemite. I thought there might be an IP address conflict so I reset it. However I still get another device's name (someone else's iPhone, to be exact) when I use the hostname command:  Last login: Fri May  1 15:52:42 on ttys000 SomeonesiPhone:~ irem$  SomeonesiPhone:~ irem$ hostname SomeonesiPhone.wireless.duke.local   What's going on and how can I fix this?  It appears the IP address has not changed. But the address that I get by doing ipconfig getifaddr en0 is different from my public IP address.  I looked up my public IP address using Google, like this: In Uzbekistan I was stopped, searched and questioned six times in one day.  At the border with Tajikistan, one official tried hiding my paperwork to elicit something from me.  In Kyrgyzstan, I was physically harassed and detained by the police, and about US$30 taken while they searched everything on me and a Dutch backpacker I was going to get food with.  I'm not particularly dodgy looking - it's happening to a lot of backpackers I meet.  Tourists are apparently a prime target for corrupt officials.  Any suggestions or tactics for avoiding these occurrences?  Aside from carrying only photocopies of your passport so that they can't hold your passport ransom, I've not heard many successful tips... I've been perusing the internet looking for interesting problems to solve. I found the following problem and have been going at it for the past 30 minutes with no success:     Find a function $f: \mathbb{R} \to \mathbb{R}$ satisfying $f(f(x)) = -x$ for all $x \in \mathbb{R}$.   I am also wondering, can we find $f$ so that is continuous?  I was thinking of letting $f$ be a periodic function, and adding half the period to x each time. I had no success with this, and am now thinking that such a function does not exist.  Source: Tips for getting really good answers on SO The way the askers write questions is a factor in determining how good the answers we get are. Obviously, if you ask a clear, direct question it is easier for a someone to answer you.  But, sometimes I've done that and my questions have still been overlooked.  I've noticed some question-posting methods that aren't exactly obvious, but do help me get better answers.    Some tricks I've learned...   Even if my question is very complicated, I try to write a short question.  People get turned off by longer questions, probably because they think its not worth their time (relative to other questions with equal reputation potential).  After I get some traffic and votes on my question, and maybe some comments like "Can you post the code?" or "Clarify this..." I might post more, but by then I am already getting attention to my question.  So, in summary, even if I know the answer requires a long question, I don't make it long until I've already gotten people to look at it. I use more generic tags when possible.  People are more likely to have c# and linq as a favorite tag, than MethodCallExpression, and I tend to get more views if I use at least one or two general purpose tags along with my more specific tags. I've noticed that if I give my question the performance tag, then SO heavy-weights like Marc Gravell, Jon Skeet and Eric Lippert are more likely to answer my question (especially if I use it in conjunction with a C# tag).  Since having this realization, if I can think of a way to rephrase my question to sound like it is related to C# performance, I will word it as such and tag it accordingly.   "Be polite. SO Users HATE when people be impatient, rude or impolite." - Benny   So,  Does anyone else have any tips for getting better answers and question-recognition on SO?  I think it would be really helpful if everyone contributed their own observations and techniques here.  P.S. This question is probably too long. consider $r&gt;0 , p&gt;1$ and $K \subset B(x_0 , 2r) \subset R^n$ . $K$ compact. Define the sets :  $$A = \{ u \in C^{\infty}_{0} (B(x_0 , 2r)); \textit{ such  that }  \  u=1 \textit{ in a open  neighborhood of K} \}$$  $$ B =  \{  u \in  H^{1,p}_{0} (B(x_0 , 2r)) \cap  C (B(x_0 , 2r))  : u=1 \  on \  K  ,  \  0 \leq u \leq 1\}$$  I want to show that   $$ \displaystyle\inf_{u \in A} \displaystyle\int_{B(x_0 , 2r)} |\nabla u|^p  \ dx= \displaystyle\inf_{u \in B} \displaystyle\int_{B(x_0 , 2r)} |\nabla u|^p   \ dx$$  For this i am trying to aproximate (in the sense of the  norm of $H^{1,p}$) the function of B , by function of $A$.  Somenone can give me a hint ?  thanks in advance Find the value of $a^{2m}+a^m+{1\over a^m}+{1\over a^{2m}}$ Let $a$ be a complex number such that $a^2+a+{1\over a}+{1\over a^2}+1=0$  Let $m$ be a positive integer, find the value of $a^{2m}+a^m+{1\over a^m}+{1\over a^{2m}}$  My approach: I factorized equation 1 which yielded $(a+{1\over a})={-1^+_-(5^{1\over2})\over2}$  I don't know if it is of any help. Should initial e's after h-dropping be capitalized? A new Victorian-set British novel begins sentences containing h-dropping with lower-case e's, as in " 'e took my money, 'e did." This seems incorrect but I'm at a loss to find the rule. What is the difference between implication symbols: $\rightarrow$ and $\Rightarrow$? If $p$ is a prime and $p \mid ab$, then $p \mid a$ or $p \mid b$. I am very new to cryptography (so be kind), but I have a question that may seem silly.   If the one-time pad is the perfect cipher and impossible to crack, why would the following algorithm not be one of the strongest:  To encrypt:   generate a random bit array using a pseudo number generator with the key as the seed  The generated bit array is the size of the data use a simple XOR to encrypt using the generated bit array   To decrypt:   pseudo-random number generator will generate the same bit array with the same seed (the key) generate bit array the same size of the encrypted data run XOR on encrypted data using bit array that's generated   This is very simplified, but you can imagine how for larger encryption you could break down the data into blocks, and in order to ensure that the block bit arrays are random a new seed for each can be derived from the key. So, if $f(k)$ is the pseudo random number generator, and b is the block number, you can encrypt each block with $f(k+b)$. ($k$ is key)   Instead of using a pseudo-random number generator, you could use some hash algorithm to generate the pad.    (assuming you can use equivalently strong keys) would this not be a very strong cipher method? Perhaps it may be strong and heavy on memory since the pad needs to be the size of the data. So during encryption and decryption you need to generate a pad that's equivalent to the size of the data.   Is there a flaw in this algorithm, or would something like this not be practical and much slower than current standards? How do I reverse a list using recursion in Python? I want to have a function that will return the reverse of a list that it is given -- using recursion. How can I do that?