Can someone guide me on how to find solution to such problems within a minute as that is the amount of time I will be given during my exams. also share what answer you get as I got different answers for same question with different methods
How do I efficiently compute $a^b\bmod c$:  When $b$ is huge, for instance $5^{844325}\bmod 21$? When $b$ is less than $c$ but it would still be a lot of work to multiply $a$ by itself $b$ times, for instance $5^{69}\bmod 101$? When $(a,c)\ne1$, for instance $6^{103}\bmod 14$?  Are there any other tricks for evaluating exponents in modular arithmetic?  This is being asked in an effort to cut down on duplicates, see  and .
How do you make a torus 'telescope' open? Remove numbering from Section and Subsection Why does evolution not make our life longer? Why does evolution not make life longer for humans or any other species?  Wouldn't evolution favour a long life? How do constant aperture zoom lenses work? How to fix "Duplicate sources.list entry" warning? Every time I try to do some update (sudo apt-get update) it says:   Reading package lists ... ready W: Duplicate sources.list entry http://archive.canonical.com/ubuntu/ need /partner amd64 Packages (/var/lib/apt/lists/archive.canonical.com_ubuntu_dists_precise_partner_binary-amd64_Packages) W: Duplicate sources.list entry http://archive.canonical.com/ubuntu/ need /partner i386 Packages (/var/lib/apt/lists/archive.canonical.com_ubuntu_dists_precise_partner_binary-i386_Packages) W: You have to run apt-get update to correct these problems   sources.list:   ls /etc/apt/sources.list.d/ as requested:  milton@milton-HuronRiver-Platform:~$ ls /etc/apt/sources.list.d/ emesene-team-emesene-stable-precise.list precise-partner.list precise-partner.list.save upubuntu-com-chat-precise.list upubuntu-com-chat-precise.list.save Makeindex error when ! used in \index in align environment See MWE below. The symbol ! in the \index causes an error. This is only the case in align environments (not in equation). So it seems to have to do something with amsmath. If ! is put in an \mbox there is no error. But then of course the font size is not adjusted to subindices. Curiously enough, \index{$C_{K!!}$} (two exclamation marks) doesn't cause any error.    \documentclass{article} \usepackage{amsmath} \usepackage{imakeidx} \makeindex \begin{document} \begin{align} y\index{$C_{K!}$} \end{align} \printindex \end{document}     Any solutions suggested? What is reflection, and why is it useful?  I'm particularly interested in Java, but I assume the principles are the same in any language. How to repair Ubuntu installation after interrupted dist-upgrade without losing any custom settings? U.K visitor visa with very old criminal conviction - do I disclose it? I have been travelling to the U.K for 25 years at least 8-10 times a year,  receiving 5 year visas, never breaking the stay period, with no problem. I've gone through immigration with no problem till my last time in London. I was stopped for eight hours and questioned about a run with the law while I was in college. This was in the US, I was around 20, it was for a credit card use. I received a conditional discharge / probation conviction which was discharged one year later. It was stupid, I was young and in college, more than 28 years ago.  After being questioned about the issue in the US, I was allowed in. They were very polite and professional.  I was also asked if I ever changed my name: I did not change it, but one letter was added in the middle of my first and last names around 20 years ago as our names are translated from Arabic to English and before the digital biometric passports, hand written passports some times varied. It was a common thing at the time.  Now I am applying for my new 5 year visas and I intend to answer truthfully  about previous conviction (previously I answered no because my conviction was spent and I expunged my record.) I want to answer yes and explain it. Is that the right thing to do?  Should I answer yes to the question about having a different name even though it was something as simple as spelling Bryan instead of Brian or Teresa and Theresa -- not really changing it and never thought of it as a change.  I have an impeccable credit rating in the Uk and the last 28 years have never received a moving violation or ever had a run in with the law.  I made one mistake when I was very young but I am  very worried about how to go ahead. I believe honesty is the best approach, is it? I have posted a proof below, and would appreciate it if someone could review it for accuracy. Thanks!   Problem:   Let n $\in$ $\mathbb{Z}$ with $n$ $\ge$ 3. Prove the following:   (a) Z(D$_{2n}$) = 1 if $n$ is odd.   (b) Z(D$_{2n}$) = {1, r$^k$} if $n$ = $2k$.   Note that $r$ and $s$ generate D$_{2n}$ with the group presentation { $r$,$s$ | $r$$^n$ = $s$$^2$ = 1, $rs$ = $sr$$^{-1}$ }   Proof:  part (a)   Let $n$ $\ge$ 3 where $n$ $\in$ $\mathbb{Z}$ and $n$ is odd.   For any x $\in$ Z(D$_{2n}$), x must commute with both $s$ and $r$, since both are in $D_{2n}$. Note that if $x$ commutes with both $s$ an $r$, then $x$ commutes with any element of $D_{2n}$ since $r$ and $s$ generate D$_{2n}$.    Then for any x$\in$ Z(D$_{2n}$), we have $xr$ = $rx$, where x is of the form $x$ = s$^j$r$^w$ (any such element can be arranged into this form via the relation $rs$ = $sr$$^{-1}$ and the fact that $r$ and $s$ generate D$_{2n}$). Note that since r and s have finite order, the exponents are modulo n and modulo 2 for $r$ and $s$ respectively.   Then we have  s$^j$r$^w$$r$ = $r$s$^j$r$^w$.   Which implies s$^j$r$^{w+1}$ = s$^j$r$^{w +- 1}$  In the case where the power of r on the RHS is $w$ - 1, it is clear we cannot have equality unless |r| = 1, which it is not. Note that if $j$ is even then we have only the $w$ + 1 case. Hence $x$ may be of the form $1$$r$$^w$ = $r$$^w$.   Also we have that x must commute with $s$, hence $s$$x$ = $x$$s$.   Then $s$$r$$^w$ = $r$$^w$$s$.  Which implies $s$$r$$^w$ = $s$$r$$^{-w}$.   Applying s$^{-1}$ to both sides we arrive at the task of finding when r$^w$ = r$^{-w}$ i.e when $w$ = $-w$. But since r has finite order we have:   $$ \bar{w} = (n-1)*w $$   where $\bar{w}$ is the residue class of w modulo n. Hence for some a $\in$ $\mathbb{Z}$$_+$   $$ w + an = wn - w $$  $$ 2w = wn - an $$ $$ 2w = n(w-a) $$   Now since 0 $\lt$ w $\le$ n we have that the LHS is greater than 0. Also since n $\ge$ w then (w-a) $\le$ 2, since otherwise the RHS would be greater than the LHS. So since n is odd we must have that (w-a) is even and hence (w-a) = 2, implying that 2w = 2n and hence n = w.   Hence x = $r$$^w$ = $r$$^n$ = 1. So Z(D$_{2n}$) = 1.   part (b)   From the argument above any x $\in$ Z(D$_{2n}$) must be of the form $s$$^j$$r$$^w$ for j even and j $\in$ $\mathbb{Z}$. Then x is of the form $r$$^w$ for $0$ $\lt$ w $\le$ $n$. Since $n$ is even we have the equation below is satisfied when $2w$ = $n$ or $w$ = $n$, since (w-a) $\le$ 2 but also (w-a) $\ge$ 0 for (w-a) $\in$ $\mathbb{Z}$$_+$.   $$ 2w = n(w - a) $$   Hence $r$$^w$ $\in$ Z(D$_{2n}$) when $2w$ = $n$, as desired. poll() failed, context incorrect? - Example: bpy.ops.view3d.background_image_add() conflict between hyperref and enumitem packages (and \refstepcounter) The solution outlined for each of the following two posts appears to conflict with the hyperref package.         Here is a MWE.  The desired output for the example is  1. Outer 2.1. This is line 2.1, in block 2.  2.2. Inner again 3. Outer again 4. Let’s end here   This output is indeed produced by the LaTeX source below, but if I uncomment the \usepackage{hyperref} line in the header, LaTeX gives me an error just after the \skipitem command, as described in the inline comment below:  \documentclass{article}  % \usepackage{hyperref} % if we uncomment the above line, we get an error as described below  \usepackage{enumitem}  \newlist{steps}{enumerate}{9} \setlist[steps]{itemindent=0pt} \setlist[steps,1]{label=\arabic*.,ref=\arabic*} \setlist[steps,2]{label=\arabic{stepsi}.\arabic*.,ref=\arabic{stepsi}.\arabic*}  \makeatletter \newcommand{\skipitem}{\refstepcounter{\@enumctr}} \makeatother  \begin{document} \begin{steps} \item Outer   \skipitem   % If we uncomment the \usepackage{hyperref} line in the header,   % we get the following error after the \skipitem above   %    % ./tmp.tex:20: Missing number, treated as zero.   % &lt;to be read again&gt;    %   \c@0.0ptxcii    % l.20   \skipitem   % \item[] \label{blockA}   \label{blockA}\item[]   \begin{steps}   \item\label{lineB} This is line~\ref{lineB}, in block~\ref{blockA}.   \item Inner again   \end{steps} \item Outer again \item Let's end here \end{steps}  \end{document}   Can someone help me work around this conflict, so I can use hyperref with this use of enumitems and refstepcounter?  Thanks.  (Note: smaller working examples are possible, but they give different error messages.  I'm hoping to get the current example working.) $$\sum_{n = 1}^\infty \frac{(-1)^n}{p_n}$$ where $p_n$ is the $n$th prime.  I have computed this to 10000 rather than infinity. My results suggest that convergence does happen but it's very slow. But I can't even be sure about the first few digits: $-0.26959$?  I have looked at the "questions that may already have your answer" and some of the "similar questions," but they involve somewhat different formulas. Finding additions and updates between two vector layers using QGIS I'm trying to find the name of a lower-budget science fiction film that was available for viewing on demands on MIT student cable in the late 1990s.  I suspect the film was made earlier in that decade, and it definitely seemed to have been influenced by RoboCop.  The plot involves characters is a somewhat dystopian near future.  One of the characters (named Hawkins, I think) is initially in a wheelchair, and that is interfering with his relationship with the woman he loves.  (He watches virtual reality simulations of himself having sex with her when he's alone at home.)  Maybe halfway through the movie, he decides there's no hope for their relationship, so he lets the bad guys turn him into a RoboCop-like cyborg.  I think he ends up fighting the other main characters, but I don't remember how the movie ends.  The film was clearly shot cheaply but competently.  The color palette was rather drab, I think, and there were few indoor sets.  It's possible that the film never got a normal release of any kind, since I don't know where the students at MIT cable got ahold of it.  So does anybody know what this film was? How do I tell if ACR will work with my camera and how do I get the correct version for Photoshop? I am a beginner photographer with a Canon 7d and MacBook Pro. I'm not sure if Adobe camera Raw is compatible with either and really need advice on how to go about downloading the correct version. Is there an online community to find travel partners (for mountains, sightseeing etc.)? Is there an online community to find travel partners for mountain trekking or sightseeing, or anything else?  I don't want to travel to foreign mountains alone, but I don't know where to search for company. There's a great community for accommodation (I mean CouchSurfing) so I think there should be something in "reverse" direction (searching not the place to stay but the people to go with). Sum of a power series $n x^n$ I would like to know: How come that  $$\sum_{n=1}^\infty n x^n=\frac{x}{(x-1)^2}$$  Why isn't it infinity? Are Playstation 4 games locked to a certain region? This question has been asked before, but it was for PS3 and Xbox 360 and was asked August 2010. This time around my question goes specifically to PS4, since I'm moving from Mexico to Europe and I was wondering if the European games will run in my console or if I'll have to buy a new one. I never thought of this before (which is why I never paid attention if it was mentioned in reports or specs of the console) and the research I did wasn't too helpful. Point vs Pixel: What is the difference? What is the difference between points and pixels?