Suppose n is a product of distinct odd primes. Show $a^{\phi(n)+1}\equiv a $ (mod $n$).
Proof of $a^{\phi (n) +1} \equiv a \pmod{\! n} $ Carmichael generalization of Fermat-Euler's theorem.
I am facing problem to enable wifi on my Lenovo G50-30 Below are the command out put please help me out   lspci | grep -i wireless:  02:00.0 Network controller: Realtek Semiconductor Co., Ltd. RTL8723BE PCIe Wireless Network Adapter     nm-tool:  NetworkManager Tool  State: disconnected  - Device: eth0    Type:              Wired   Driver:            r8169   State:             unavailable   Default:           no   HW Address:        28:D2:44:81:36:0B    Capabilities:     Carrier Detect:  yes    Wired Properties     Carrier:         off   - Device: wlan0   Type:              802.11 WiFi   Driver:            rtl8723be   State:             unavailable   Default:           no   HW Address:        14:2D:27:21:8D:47    Capabilities:    Wireless Properties     WEP Encryption:  yes     WPA Encryption:  yes     WPA2 Encryption: yes    Wireless Access Points      nmcli nm:  RUNNING         STATE           WIFI-HARDWARE   WIFI       WWAN-HARDWARE   WWAN       running         disconnected    disabled        disabled   enabled         disabled       sudo rfkill list:  0: ideapad_wlan: Wireless LAN     Soft blocked: no     Hard blocked: yes 1: ideapad_bluetooth: Bluetooth     Soft blocked: no     Hard blocked: yes 2: hci0: Bluetooth     Soft blocked: no     Hard blocked: no 3: phy0: Wireless LAN     Soft blocked: no     Hard blocked: no How to fit a tight ellipse around nodes I have a set of two nodes and I would like to draw tight ellipses around them in my tikz picture. Note that this is part of a beamer presentation.    I have attached an image that illustrates the wanted effect.    Here is what I have so far:  \documentclass[10pt,aspectratio=1610]{beamer} \usepackage{tikz} \usetikzlibrary{shapes,arrows,backgrounds,fit,positioning}  \begin{document}  \tikzset{res/.style={ellipse,draw,minimum height=0.5cm,minimum width=0.8cm}} \tikzset{literal/.style={rectangle,draw,minimum height=0.5cm,minimum width=0.8cm,text width = 1.2 cm, align = center}}  \begin{frame}[fragile]{Example}{} \begin{tikzpicture}[node distance = 2 cm,auto, ] \node[res,] (obs1) {\footnotesize :obs1} ; \node[res, left = 1cm of obs1] (shop1) {\footnotesize :shop1} ; \node[literal, left = 1cm of shop1] (shopName1) {\footnotesize ``Barnes \&amp; Noble''} ; \node[res, above of = shop1] (book1) {\footnotesize :book1} ; \node[literal, left = 1cm of book1] (bookName1) {\footnotesize ``Dune''} ; \node[res, below of = obs1] (obs2) {\footnotesize :obs2} ; \node[res, left = 1cm of obs2] (shop2) {\footnotesize :shop2} ; \node[literal, left = 1cm of shop2] (shopName2) {\footnotesize ``Amazon''} ; \node[res, below of = shop2] (book2) {\footnotesize :book2} ; \node[literal, left = 1cm of book2] (bookName2) {\footnotesize ``Dracula''} ; \node[literal, below of = obs2] (measure2) {\footnotesize 12} ; \node[literal, above of = obs1] (measure1) {\footnotesize 7} ; \node[literal, right = 1cm of obs1] (monthName) {\footnotesize ``January''} ; \node[literal, right = 1cm of measure1] (yearNum) {\footnotesize ``2016''} ; \node[res, right = 1cm of monthName] (month) {\footnotesize :january2016} ; \node[res, above of = month] (year) {\footnotesize :2016} ; \node[res, below of = month] (day) {\footnotesize :date31012016} ;  \path[-&gt;,draw]     (obs1) edge node[midway,above=-6pt,fill=white,inner sep=0pt] {\footnotesize :shop} (shop1)     (obs2) edge node[midway,above=-6pt,fill=white,inner sep=0pt] {\footnotesize :shop} (shop2)      (obs1) edge node[midway,above=-6pt,fill=white] {\footnotesize :book} (book1)     (obs2) edge node[midway,above=-6pt,fill=white] {\footnotesize :book} (book2)      (book1) edge node[midway,above=-6pt,fill=white,inner sep=0pt] {\footnotesize :name} (bookName1)     (book2) edge node[midway,above=-6pt,fill=white,inner sep=0pt] {\footnotesize :name} (bookName2)      (shop1) edge node[midway,above=-6pt,fill=white] (title1) {\footnotesize :title} (shopName1)     (shop2) edge node[midway,above=-6pt,fill=white] (title2) {\footnotesize :title} (shopName2)      (obs1) edge node[midway,above=-6pt,fill=white] {\footnotesize :price} (measure1)     (obs2) edge node[midway,above=-6pt,fill=white] {\footnotesize :price} (measure2)      (obs1) edge node[midway,above=-6pt,fill=white]  {\footnotesize :date} (day)     (obs2) edge node[midway,above=-6pt,fill=white] {\footnotesize :date} (day)      (day) edge node[midway,above=-6pt,fill=white] {\footnotesize skos:broader} (month)     (month) edge node[midway,above=-6pt,fill=white] {\footnotesize skos:broader} (year)      (month) edge node[midway,above=-6pt,fill=white,inner sep=0pt] {\footnotesize :month} (monthName)     (year) edge node[midway,above=-6pt,fill=white] {\footnotesize :year} (yearNum)     ;  \begin{pgfonlayer}{background}   \node[fit=(shopName1)(shop1), fill=violet!30,ellipse] {};   \node[fit=(shopName2)(shop2), fill=violet!30,ellipse] {};   \node[fit=(bookName2)(book2), fill=violet!30,ellipse] {};   \node[fit=(bookName1)(book1), fill=violet!30,ellipse] {};   \node[fit=(measure1)(obs1), fill=blue!30,ellipse] {};   \node[fit=(measure2)(obs2), fill=blue!30,ellipse] {};   \node[fit=(year)(yearNum), fill=green!30,ellipse] {};   \node[fit=(month)(monthName), fill=green!30,ellipse] {}; \end{pgfonlayer} \end{tikzpicture} \end{frame} \end{document}   Here is the result that it produce    Note how the ellipses do not fit the nodes but cover much more. How can I create a tight ellipse around a pair of nodes? My question might appear the same as the question , but actually I did read all the answers and none of them helped me with the details I need.  I am new to cryptography and my problem is with two time pad attacks on OTP. The problem I had in my course was that I have 10 ciphertexts encrypted with the same key $K$. I am then given another ciphertext that I should decrypt.  I know that XOR-ing two ciphers gives me the XOR of their original messages.  My question is what is the correct thing to do after that?  I tried to take 3 ciphertexts $C_1, C_2$ and $C_3$.  Then get $S_1 = C_1 \oplus C_2  \oplus $' ', also get $S_2 = C_1 \oplus C_3 \oplus$ ' '.  After that I compared all corresponding characters in $S_1$ and $S_2$,  and if $S_1[i] = S_2[i]$ then I calculate $S_1[i] \oplus C_2[i]$ to get $K[i]$.  I tried this on paper before coding and it worked, but I might be missing something.  Is this the right approach? Why does it work? 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