$a,b\in\mathbb{N}$ and $ab&gt;2$.Suppose,$\text{lcm}(a,b)=L,\gcd(a,b)=G$ and $a+b\mid L+G$.Prove that $\dfrac{(a+b)}{4}\cdot(a+b)\ge (L+G)$. Also prove that equality occurs when $a,b$ are consecutive odd integers.    Hint: There is an exception when $a,b$ are both equal to 2.  I could not approach this problem at all. Please help.
Let $a$ and $b$ be natural numbers with $ab&gt;2$. Suppose that the sum of their highest common factor and least common multiple is divisible by $a+b$. Prove that the quotient is at most $\frac{a+b}4$. When is this quotient exactly equal to $\frac{a+b}4$.
I have a small question regarding using sudo with output redirect &gt;. To enable IP forwrding, someone can use the command:  echo 1 &gt; /proc/sys/net/ipv4/ip_forward   Executing this command will give permission denied as it requires root privileges. However, executing the same command with sudo gives also permission denied error! It seems that output redirect &gt; does not inherit the permissions of the preceding command echo. Is this right?  As a workaround I do:  echo 1 | sudo tee /proc/sys/net/ipv4/ip_forward   Is this the best way to do it? Am I missing something?  Please, note that this is an example and it applies to all commands that use output redirect. Word for a person who knows two or three languages A polyglot is a person who is fluent in many languages, but what do we call a person who is fluent in only two or three languages?  Is bi-glot a proper term for this?  I don't think the words bilingual / trilingual fit the bill, because I am looking for a -glot word. Elegant way of obtaining the envelope of oscillating function Old phone cannot connect to computer, the "computer connection" icon appears on phone I would like to get some pictures and audio out of an AT&amp;T SGH-a687 phone onto my Mac computer. I first tried connecting it to a New PC Studio on a PC, but there was a problem after transfer 2 photos, so I tried again. But the PC could no longer detect the phone, even though the "connected to computer" icon appears on the phone. I tried a different cable, different USB ports. No use.  So I downloaded the Samsung Kies on my Mac. Same thing, it tries to search for the phone once I connect the phone, but the computer doesn't detect anything. The phone still shows the "connected to computer" icon.  There is also Bluetooth (no internet on the phone) on the phone, but it failed to send the picture each time I tried.  What can I do at this point? Please also suggest data extraction software for either Mac or PC, if that's the easier route. Effective Refractive Index in Paper I am reading the paper, entitled "Artificial uniaxial and biaxial dielectrics with use of two-dimensional subwavelength binary gratings" by Eric B. Grann, M. G. Moharam, and Drew A. Pommet.   In the paper they mention this formula:        Do you know how it can be derived and where it comes from ? I need help with this:  If a is an element of $\mathbb Z_n$ and $\gcd(a,n) &gt; 1$ , then a is not invertible.   First you show that if $a$ is an element of $\mathbb Z_n$ and $\gcd(a,n)&gt;1$, then there  is an element $b$ of $\mathbb Z_n$ and ($b$ is not equal to the zero element) from which $ab = 0$.   The second part I cant get. i.e. show if $b$ is not the zero element  and $ab=0$ then $a$ is not invertible. Sophomore's dream: $\int_0^{1} x^{-x} \; dx = \sum_{n=1}^\infty n^{-n}$ How was this answer posted after this question was closed? What is the volatile keyword useful for? How does MySQL store data Find out which application listens to specific key bindings? Is there an application to know which key is bound by which process and therefore not usable in another application? Which is the best way to install new hplip versions? Hplip is the software that drives most hp printers. Usually new printers are only supported in newer versions of the software. Canonical doesn't backport new versions packages to older Ubuntu versions, that means that newer drivers are not going to be available in the Software Center of older Ubuntu versions.  As an example yesterday I needed to hook a hp 1005p to a computer running 8.04. I was forced to install a newer hplip than the version available in the repositories. I installed it using the script from the hp page, but I'd like to know other alternatives.  Which is the best (technically, not the easiest) way to install it and why?. How do I completely remove Ubuntu? Cortana Search is not finding applications on Windows 10 For the past several versions (since Windows Vista, I think), if you hit the Windows key and start typing, Windows will search for applications.  Since upgrading to Windows 10 with , she is only hit or miss at finding applications. And she doesn't do partial searches, either.  Some examples:   WinKey + type "Paint" does not find "mspaint". Instead it suggests some applications from the store. WinKey + type "Excel" does not find Excel. Nor does typing "Word" find Word. However, typing OneNote finds OneNote.   What is the reason for this behavior and how I can get Cortana to find the applications installed on my machine again? Is there a setting I am missing?    If I browse to "C:\ProgramData\Microsoft\Windows\Start Menu\Programs" I can see that the shortcuts for these applications all still exist. But Cortana is not finding them. Everything from the Microsoft Office applications to  to  to hundreds of other applications I have installed.  Where can I set which folders Cortana indexes?    As you can see in the screen shot below, this Start Menu folder is, indeed, indexed:    However, none of these applications are coming back as search results from the "start" menu.  It may be related to  as well.  I did a brand new install of Skype (for desktop), and it shows up nowhere in the start menu. And typing in "Skype" to search for the application does not find it. But the shortcut to Skype is definitely in this location:  C:\Users\Patrick\AppData\Roaming\Microsoft\Windows\Start Menu\Programs\Skype     EDIT  The latest Windows 10 update (Threshold 2) has solved this issue for me. Cortana now finds all my files with no issues. Previously my only solution was to use , I've been able to uninstall that now and use the standard Windows Start menu. Comparing and contrasting, p-values, significance levels and type I error Is there a simple way in Perl that will allow me to determine if a given variable is numeric?  Something along the lines of:  if (is_number($x)) { ... }   would be ideal.  A technique that won't throw warnings when the -w switch is being used is certainly preferred. ${!FOO} performs a double substitution in bash, meaning it takes the (string) value of FOO and uses it as a variable name. zsh doesn’t support this feature.  Is there a way to make this work the same in bash and zsh?  Background:  I’ve got a list of environment variables, like  PATH MAIL EDITOR   and want to first print the variable names and afterwards their values.  This works in bash but not zsh:    for VAR in LIST do         echo $VAR         echo ${!VAR} done   It should be somehow possible “the old way” with eval, but I can’t get it to work:  for VAR in LIST do         echo $VAR         echo `eval \$$VAR` done   I’m never going to understand why I can’t simply do arbitrary deep substitutions like ${${VAR}} or even ${${${VAR}}} if need be, so an explanation for that would be nice, too. Prove: If a sequence converges, then every subsequence converges to the same limit. I need some help understanding this proof:  Prove: If a sequence converges, then every subsequence converges to the same limit.  Proof:  Let $s_{n_k}$ denote a subsequence of $s_n$. Note that $n_k \geq k$ for all $k$. This easy to prove by induction: in fact, $n_1 \geq 1$ and $n_k \geq k$ implies $n_{k+1} &gt; n_k \geq k$ and hence $n_{k+1} \geq k+1$.  Let $\lim s_n = s$ and let $\epsilon &gt; 0$. There exists $N$ so that $n&gt;N$ implies $|s_n - s| &lt; \epsilon$. Now $k &gt; N \implies n_k &gt; N \implies |s_{n_k} - s| &lt; \epsilon$.   Therefore: $\lim_{k \to \infty} s_{n_k} = s$.   What is the intuition that each subsequence will converge to the same limit I do not understand the induction that claims $n_k \geq k$ How do I change the default file manager back to Nautilus? Mixing and Distributing Qualifiers ($\forall x$, $\exists x$)