Last two digit of number raised to exponents. Find the last two digits of $3^{3^{2014}}$.   Attempt:   First I try to work with $3^{2014}$. So we can work on $\text{mod 10}$.  Then, $$\begin{align}&amp;3^1 \equiv 3\pmod{10}\\ &amp;3^2 \equiv 9 \pmod{10}\\ &amp;3^3 \equiv 27 \equiv 7\pmod{10}\\ &amp;3^4 \equiv 81 \equiv 1\pmod{10}\end{align}$$  So  $3^{2014} = 3^{(4\cdot503) + 2} \equiv 3^0 \cdot 3^2 \equiv 9 \pmod{10}$.  So the last digit is 9. However I am ask to find the last two digit. Can anyone please help me? I would really appreciate all the help.
Last digits, numbers Can anyone please help me?  1) Find the last digit of $7^{12345}$   2) Find the last 2 digits of $3^{3^{2014}}$.  Attempt: 1)  By just setting the powers of $7$ we have $7^1 = 7$, $7^2=49$, $7^3=343$, $7^4 = 2401$, $7^5 = 16807$, $7^6 = 117649$, $\dots$  After the power of $4$, the last digits will repeat. Then by noticing the pattern the digits will end in $7,9,3$ and $1$.  Then we can divide the exponent $(12345)$ by $4$ since this is the cycle that makes it repeat. Then $12345 : 4$ has remainder $1$. So $7^1 = 7$ is the unit digit to $7^{12345}$.   So the last digit is $7$.   I know how to do it like this, the problem does not state how to find the last digit, but I know it has something do do with Euler's theorem.  for part $2$) I don't know how to start. Can anyone please help me?   Thank you for the help.
Idiom for a speech with redundant information Should actors in a game be responsible for drawing themselves? I am very new to game development, but not to programming.  I am (again) playing around with a Pong type game using JavaScript's canvas element.  I have created a Paddle object which has the following properties...   width height x y colour   I also have a Pong object which has properties such as...   width height backgroundColour draw().   The draw() method currently is resetting the canvas and that is where a question came up.  Should the Paddle object have a draw() method responsible for its drawing, or should the draw() of the Pong object be responsible for drawing its actors (I assume that is the correct term, please correct me if I'm incorrect).  I figured that it would be advantagous for the Paddle to draw itself, as I instantiate two objects, Player and Enemy. If it were not in the Pong's draw(), I'd need to write similar code twice.  What is the best practice here?  Thanks. How does vampirism work? A PC in our group was off on his own following a recurring villain of our groups. He was paralyzed where he could move, and the vampire bit his neck. The vampire said "now you get to remain 19 (years old) forever" before he left the cave. This was all done in RP.  We found him later, and healed up his bite marks. Our characters know of vampires, specifically only the one that bit him (recurring villain), but don't really know how vampirism works, because it isn't a super common thing in this world yet.   My question as the player is:  How does RAW vampirism work in 5e?  Is it a disease that develops over time? Is it something where they have to die and be brought back as a vampire? Selected references without hyperlink When using the hyperref package, is it possible to tell LaTeX not to create a link for some references?  In my case the page number is displayed in the footer in  \fancyhf[FC]{\thepage\ of \pageref{pg:end}}   which is what I want. Unfortunately every single page in the document then has a link to the final page, which is a bit silly.  Any ideas on how to fix that? Generator expressions vs. list comprehensions When should you use generator expressions and when should you use list comprehensions in Python?  # Generator expression (x*2 for x in range(256))  # List comprehension [x*2 for x in range(256)] "going to" vs "will" I know several questions were asked about the difference between "going to" and "will". Based on several answers (see, for instance, ,  and ), I understood that "will" is more spontaneous and "going to" is used with more planned actions.  So, it seems that everything is pretty fine. However, in , Kosmonaut has an answer in which he states: "Let's say that tomorrow you will walk your dog from 7 - 8 AM".  On the one hand, you probably planned to walk your dog long before and thus I should use "Let's say that tomorrow you're going to walk your dog from 7 - 8 am".   On the other hand, since I'm saying "let's say...", I'm deciding right now (thus, unplanned) that you will walk your dog. So, even though in this hypothetical situation you made a plan, I'm in a more spontaneous mood deciding right now that that's what you will do tomorrow, and, thus, I should use "will".  Which one (if any) of the above explanations is right? What would have happened if Voldemort had won Priori Incantatem before Nagini had been decapitated? As wielder of an elder wand that belonged to Harry, during Voldemort’s Last Stand, there are two instances of Priori Incantatem occurring, back to back.  Basically, the current order of events makes total sense. Since Nagini was the last horcrux, when the body died, the fragment of Voldemort’s soul locked in it was released. This disrupted the first instance of Priori Incantatem, and Voldemort was now significantly weaker than Harry – so Harry won. Naturally, the Elder Wand’s killing spell was reflected back on Voldemort because of Priori Incantatem outcome.   But, if Voldemort was just told by Harry that the Elder Wand wouldn’t work against Harry because it belonged to Harry, and was a wand that would win any duel for its master, why would Voldemort willingly try and continue to use it? This is doubly so, since after he had cast the killing spell on Harry once it had only torn Voldemort’s soul from Harry’s body – wouldn’t that even imply that the Elder Wand would realistically refuse almost any attempted direct harm against its owner?  Also, wouldn’t a first refusal of the Elder Wand to kill it’s own master also imply an additional refusal even if Voldemort had won either of the Priori Incantatem matches (in which case, the worst that could happen in this line of thinking is, Voldemort has Harry’s wand since Harry’s spell is Expelliarmus)? There is a book I read in the 1980s or 1990s from a Ca.  library. It was a hard copy not a paperback. It is not "To Outlive Eternity/Tau Zero" by Poul Anderson (I have that one). It may have been written by a woman or someone whose last name starts with an "L".  A large crew of men and women are in a new super fast space ship. They are being chased by someone and they outrun the collapse of the universe, but they have lost their home planet. They watch as the universe is reborn. They go into orbit around a planet that looks like it can support life like theirs. They are moving super fast. They look like a ring of radiation going around the planet. They put down 10 super metal markers with scientific info onto the surface of the planet. They watch the planet develop and form life like the dinosaurs and then evolve into human like life forms.  There are 2 warring countries on the planet. They block or turn off the sun to freeze the planet and force the countries to form one world government. The planet gets freezing cold and the people start to freeze. The countries agree to form one world government. The people on the ship are welcomed to the planet. Cannot see OpenLayers under Plugin in QGIS? Although I have installed the OpenLayers plugin in QGIS 2.4, I cannot see it in the list when I click on Plugins.    Does anyone else have this issue? Print lines between (and excluding) two patterns PDF video Embedding Problem I have embed an mp4 video in the PDF file as given below:  \documentclass{beamer} \usetheme{Singapore} \usepackage{media9} \usepackage{hyperref} \begin{document}  \begin{frame}\frametitle{Title}  \includemedia[ width=\paperwidth,height=0.7\linewidth, activate=pageopen, flashvars={aaa.mp4 &amp;autoPlay=true} ]{}{C:/Users/name/Desktop/aaa.mp4}  \end{frame}  \end{document}    There is no compile time error in the code but after running it this massage comes along a  blank pd f file....  "To view the Flash @R Technology content in this pd f file, please install the required version of Flash Player"  I would like to add here that I already have installed the latest version of Flash Player(Adobe Flash Player 12 ActiveX). So,Can anyone tell what version is required to be installed to obtain Video in pd f file. Insert Login IP Ranges using apex in salesforce Is it possible to insert to Login IP ranges via apex? Distance from the surface as an input node for use with volumetric materials I'd like to find the files in the current directory that contain the text "chrome".  $ find . -exec grep chrome find: missing argument to `-exec'   What am I doing wrong? Prove $(x_1+.....+x_n)^2 \leq n(x_1^2 + .....+x_n^2)$ Prove that  $(x_1+\dots+x_n)^2 \leq n(x_1^2 + \dots+x_n^2)$  for all positive integers n and all real numbers $x_1,....,x_n$  I am attempting a proof by induction but wasn't sure if i need the Cauchy-Schwarz Inequality or perhaps another way other than induction to prove this.  Proof   $n=1$ true  assume true for $n=k$  Now for $n= k+1$  $(x_1 + \dots +x_k + x_{k+1})^2 \leq \dots$ I am teaching myself information about groups, but don't really understand how to work through this problem. Here is what I have been thinking so far (please note that I do not need to work through showing any closure): $\mathbb{Z}^*$ denotes the nonzero integers. Prove or disprove: $(\mathbb{Z}^*, \cdot)$ and/or $(\mathbb{Z}^*, \div)$ is a group.  For $(\mathbb{Z}^*, \cdot)$:  Group. Associative Law: Let $a,b,c \in \mathbb{Z}^*$. Then $(a\cdot b)\cdot c=a\cdot (b \cdot c)$, and the associative law holds. Existence of Identity: Let $a \in \mathbb{Z}^*$, then $a \cdot 1 = 1 \cdot a = a$, and the identity holds. Existence of Inverses: Let $a \in \mathbb{Z}^*$. Then the inverse of $a^{-1}=-a$, and the inverse holds. Note that this only works because we are ommitting zero.  Hopefully this looks like I am on the right track. I don't know to what extent I can explain logic vs showing it with mathematical operations, so I would appreciate guidance in this respect.  For $(\mathbb{Z}^*, \div)$: Not a group. Disprove by counterexample.  Yeah, I dunno about this one. I don't think it's true, because we will be introduction rational expressions that won't allow this to hold, but I'm not sure where I can use this information to disprove the statement. Here is my attempt to prove the Well-Ordering Principle, i.e. that any non-empty subset of $\Bbb N$, the set of natural numbers, has a minimum element.  Proof. Suppose there exists a non-empty subset $S$ of $\Bbb N$ such that $S$ has NO minimum element. Define $A = \left\{n\in \Bbb N : (\forall s\in S)(n \leq s)\right\}$. It is obvious that $1\in A$. Suppose $n\in A$, then for each $s \in S$, there exists $q \in N$ such that $n + q = s$. Since $q \ge 1$, $n+1 \leq s$, for all $s\in S$. By Principle of mathematical induction, $A = \Bbb N$. Take any $s_0 ∈ S$, then $(\forall s\in S)(s_0 \leq s)$. (This contradicts that $S$ has no minimum element).  How do I prove the statement without invoking Mathematical Induction? Also, I read that the proof of Principle of Mathematical Induction makes use of Well-Ordering. Can it be proven independently of Well-Ordering too? how to add object to a keyframe range? lets assume we have an iron man pointing his hand to a target and shooting few frames later as shown in the picture below:    frame 0 to 10 he will point frame 11 to 15 he will shoot, which is just a long cube with one larger face to give idea or a laser projectile   how can I add the projectile cube only on frame 11? I don't what it to exist before that  thanks Unable to check a mdadm RAID1 array, says the file system's read only? I recently migrated a RAID1 from a CentOS 5 system to a CentOS 6 system and ever since when I attempt to perform a check I get the following:  $ echo 'check' &gt; /sys/block/md127/md/sync_action -bash: /sys/block/md127/md/sync_action: Read-only file system   This actually shows up from a CRON too, specifically, /etc/cron.d/raid-check. this is included in the mdadm RPM:  $ rpm -ql mdadm | head -5 /etc/cron.d/raid-check /etc/rc.d/init.d/mdmonitor /etc/sysconfig/raid-check /lib/udev/rules.d/63-md-raid-arrays.rules /lib/udev/rules.d/65-md-incremental.rules   Here I'm running the same command that's used by the CRON:  $ raid-check /usr/sbin/raid-check: line 96: /sys/block/md127/md/sync_action: Read-only file system   This typically runs once a week and sends an email which is what originally alerted me to the issue. But at any rate I'm at a loss why this RAID seems to be unable to be checked.  The RAID seems fine on inspection though.  $ cat /proc/mdstat  Personalities : [raid1]  md127 : active raid1 sda1[0] sdb1[1]       976759936 blocks [2/2] [UU]  unused devices: &lt;none&gt;   I point this out only because while googling I found this thread regarding a bug in mdadm but this was for a older version of mdadm.      Version info  $ lsb_release -d Description:    CentOS release 6.6 (Final)  $ rpm -q mdadm mdadm-3.3-6.el6_6.1.x86_64  $ mdadm --version mdadm - v3.3 - 3rd September 2013 $\require{enclose}\enclose{horizontalstrike}{\rm Greatest}\!$ Least prime factor of $n$ is less than square root of $n$, proof I remember reading this somewhere but I cannot locate the proof.