I know that $a \neq b$, then $\frac{a}{b}+\frac{b}{a}$ would not equal to $m$ (an integer) so I set them into one that wasn't a fraction by squaring. So I got $\frac{a^2+b^2}{ab}$. How can I show that $a^2+b^2$ is not divisible by $ab$?
Some observations I made is for $\frac{a}{b}+\frac{b}{a}$, is either:   the denominator has to be one, the numerator has to be a multiple of the denominator or  the numerator and denominator have to be the same.    Obviously, with the given conditions case 3 is eliminated since  $a \neq b$.  For case 1, if $b=1$, then $a=1$ which is a contradiction to the given condition that says $a \neq b$.  For case 2, I would think of examples. For example if $b=2$ then a multiple is $4$, so $a=4$. Then we have:  $$\frac{a}{b}+\frac{b}{a}=\frac{4}{2}+\frac{2}{4}=\frac{10}{4}$$  which is not an integer, but how would i proceed to show this case for all integers.
Proving that if $f: \mathbb{C} \to \mathbb{C} $ is a continuous function with $f^2, f^3$ analytic, then $f$ is also analytic Let $f: \mathbb{C} \to  \mathbb{C}$ be a continuous function such that $f^2$ and $f^3$ are both analytic. Prove that $f$ is also analytic.   Some ideas: At $z_0$ where $f^2$ is not $0$ , then $f^3$ and $f^2$ are analytic so $f = \frac{f^3}{f^2}$ is analytic at $z_0$ but at $z_0$ where $f^2$ is $0$, I'm not able to show that $f$ is analytic. What is a concise function that formats a (String) decimal into a currency format in Apex? Etymology of "cut someone some slack" Teenagers. All the literature tells you one thing and one thing only – that  whatever they are doing, give them a break, cut them some slack, it's normal.      From the novel, Apple Tree Yard   I'm curious about the origins of . I know slacks are American English for trousers or pants, and a slacker is a lazy person. But what is a slack and how does one "cut" it? Can you launch Remote Desktop via URL? How To Display Too Much Data We're building a web-based platform where the main dashboard shows a table of data for users to view and analyze. As we're growing this tool, we seem to be adding more and more columns and are running out of horizontal space on smaller screen resolutions (we're seeing some data wrap). It is important to the user to view all columns at once, so column hiding is somewhat out of the question, as is horizontal scrolling (per the boss's request and because horizontal scrolling sucks anyways). I was wondering if anyone knew of a clever UI for tables with many columns.  EDIT  Below is a screen capture of the layout of the table. Unfortunately for security purposes I had to blur out the column titles. This example has rows with practically no data in them, but in use they will have lots of data that will expand wider than their respective column titles. In response to some of the answers, it is imperative that the user see all the columns, as they are all necessary to analyze the data to make certain decisions. All the data is numeric aside from the first three columns. Thanks again for any feedback.    Ok so I see that uploading this image doesn't really do it justice... it takes up the full width of the screen at 1680x1050 resolution (22") and as data fills in the columns, the left column w/the text shrinks down a bit.  EDIT 2 I just wanted to say thank you to everyone's awesome ideas. I can only select one answer, but a lot of your ideas have proven very useful and may end up in the final product. Thanks again! Notify instantly when a currently viewed post is deleted At the moment SE shows us current voting results, a notification for new answers and comments, etc., instantly. However, it doesn't show us when a post is deleted.  Please notify us also when a post is deleted. Then I don't have to type a (long) comment, only to notice the post got deleted in the meantime. Excel: I need a formula that will count months between two dates. If it's 1-15 it should be a half month. Any ideas? So I am trying to find a formula for excel. Bascically I have two dates and I need to know the number of months between them. But to add complexity, I need a rule in there that if the earlier date is the 1st to 15th it will be a whole month but if it is 16th to end of the month it will be a half month.  For example: Date 1: April 4, 2012 Date 2: December 31, 2012 Answer: Number of months would be 9 months  If Date 1 was April 16, 2012, with the same Date 2 of December 31, 2012, then the number of months would equal 8.5.  Can anyone help me with this one? You help would be greatly appreciated. many thanks More examples of Simpson's Paradox, barring the ones on Wikipedia, Titanic, and delayed flights. I hope someone may know fresh examples of  for use in my statistics courses.  The examples I've been using are fine, but I'd like to have some new ones.  I'm familiar with the ones on the Wikipedia page: the gender bias lawsuit at Berkeley, batting averages, mortality rates among low birth babies, party voting on the Civil Rights Act, and success rates of kidney stone treatments.  I'm also familiar with the example of survival rates on the Titanic and the one about delayed flights at America West vs. Alaska Airlines.  These are all good ones, but, like I said, I'm looking for more. Does anybody know any others?  For those who aren't familiar with it, Simpson's paradox is essentially a property of unreduced fractions: It's possible to have, simultaneously, $$\frac{a_1}{b_1} &lt; \frac{c_1}{d_1} \text{ and } \frac{a_2}{b_2} &lt; \frac{c_2}{d_2}, \text{ but } \frac{a_1 + a_2}{b_1+b_2} &gt; \frac{c_1+c_2}{d_1+d_2}.$$   For example, $\frac{1}{3} &lt; \frac{34}{100}$ and $\frac{66}{100} &lt; \frac{2}{3}$, but $\frac{67}{103} &gt; \frac{36}{103}$.  It can be hard to spot this happening in a given real-world scenario. Using Google Music Manager in Linux from the command line My goal is to use Google Music Manager on a Linux server.  I'm wondering if the music manager can operate from the command line and with no GUI.  If I could set the software up to watch a particular directory on the server and upload from there, that would be the only necessary functionality.  Thank you. Migration stats not consistent between sites Apparently, as a result of  it appears that Yannis and myself see different statistics when going to the programmers.SE.com migration stats tool and the stackoverflow.com tool respectively.    :        while I see:        Why are the rejection rates so radically different?  Which is correct?  (Or am I simply misinterpreting the data somehow and these are representing different statistics) Definition of Significant Figures In my textbooks,  are defined as:   “Significant figures by definition are the reliable digits in a number that are known with certainty.”  “A significant figure is the one which is known to be reasonably reliable.”    Reliable means giving the same result on successive trials or reliable information can be trusted to be accurate. However, at the end of each of the two definitions it is also written that ‘the last digit of a number is generally considered uncertain in the absence of qualifying information.' For example if the mass of an object is 12.248 gm, the last digit which is ‘8’ is uncertain by plus or minus 0.001 gm. This uncertainty is unreliability in the information. So, the digit ‘8’ is uncertain, and thus according to the definition 1 it is not a significant figure. However, the rules for significant figures say that ‘non-zero digits are all significant’. Due to the last digit ‘8’, if the object was weighed with careful handling minimizing the chances of error, the figures preceding ‘8’ that are (1, 2, 2, 4) are certain and reliable. If the mass of another object is 2 gm, then it is uncertain by plus or minus 1 gm i.e. its mass could be 3 gm or could be 1 gm.  ‘2’ is therefore not certain! What is the reason that significant figures are defined to be the reliable digits in a number? In what sense they are said to be reliable? How do I handle null or optional DLL struct parameters Buy a second seat in order to bring an extra bag? Android How to use MediaScannerConnection scanFile How to create functions of arbitrary number of variables? How can I boot with an older kernel version? Proof by induction that if $n \in \mathbb N$ then it can be written as sum of different Fibonacci numbers Proof that every natural number $n\in \mathbb N$, can be written as the sum of different Fibonacci numbers between $F_2,F_3,\ldots,F_k,\ldots$.  For example: $32 = 21 + 8+3 = F_8+F_6+F_4$  Research effort:  The base step it's simple:  Let $k=1$ it can be britten as $k=1=F_2$  For the inductive step I considered:  Let $k = k  F_2 =F_2+F_2+\cdots+F_2 = \sum_{i=2}^w a_iF_i, a_i =\{0,1\} $  Then if $k$ suffice I want to see if $k+1$ suffices too... But I'm not realy seeing how to use the inductive hypothesis, so I assume It's wrong.  Any thoughts on which can the inductive step be? Can matter be described as the result of the curvature of space, instead of vice versa? Can matter be described as the result of the curvature of space, rather than the curvature of space being the result of matter, and energy being the cause of the curvature of space? is looking bad right now. No errors in the console, but the following files have no content:         The CSS on another subdomain works fine though: How do I resize my current ubuntu partition?