What is the least residue of 49^4 modulo 23 This is what I got so far but for some reason I feel like I am wrong. 49 is congruent to 26 mod(23). Therefore we have 49^4 is congruent to 26^4 = 456976 which is congruent to 456923 mod(23)but this doesnt seem like the least residue  due to the issue of the theorem stating that 0 &lt; r &lt; m but here r is bigger than m
Modular exponentiation by hand ($a^b\bmod c$) 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 .
Many older movies seem to have vivid reds and particularly realistic or noticeable skin tones (they certainly seem to stand out from typical skin tones on TV or the majority of movies). This applies even in DVD or online versions and is not, given the range of material I've noticed the effect in, an artifact of special post processing or deliberate colour manipulation.  All the movies I've noticed the effect in are from the film era, especially from when colour became common to the late 1960's. I used to think it was just Technicolor but I've seen things recently that don't claim to be Technicolor that also show the effect.   claims the effect was due to the final print being constructed late in the process for separate black and white film strips for each primary colour allowing the final dyes to be very vivid and clear:     The rich colors that the Technicolor process gave came in part from the fact that the color was not added to the process until the final stages. The color information was recorded and processed as separate black and white images which were relatively easy to control and preserve.   Is this a good explanation? In an ideal case, we assume strings and ropes to be massless and having same tension throughout.  However, I was doing some questions which involved a rope with mass.  The question was that if a uniform thick rope of length 5 m is resting on a horizontal smooth (assume frictionless) surface and is pulled by a force of 5N from one end, then what is the tension in the rope at the point 1 m away from the end where force is applied? (sorry for the poor image)    I came to the conclusion that since rope is not fixed, tension anywhere should be zero. However, when I checked the answer, it was 4 N. I failed to understand how.  Then the next part of the question was that if the rope now hangs between two fixed points which are at same level, then what would be the tension in the rope at its lowest point?    The answer has to be expressed in terms of mass of string m, acceleration due to gravity and $\theta$ where $\theta$ is the angle that tangents to the rope make with the horizontal.  I took that force acting on lowest point is mg, since it is center of mass and hence T = mg. However, according to the answer it depends upon $\theta$ as well. However, I failed to see what I was doing wrong?  I would appreciate any help in solving these questions, since I have no clue of where exactly I am wrong. I did search the Internet and a few books, but everywhere they assumed string/rope to be massless. I would like to convert a Mp4 file in HD into a Mp4 file of lower resolution. Is there any software that can be used for this? Splitting a large document into several files Restricting a user to a certain directory on SSH login I am having trouble restricting a user to a fixed directory when they connect with SSH  I want the user restricted to /var/www/home  however I connected myself and they are free to move about the system  I set the home directory to /var/www/home so they go there upon login however this doesnt restrict them.  I have tried a few things from suggestions in Google but nothing so far seems to work / do what I want  For example I have changed the chmod of www to 755 and root owned but still no effect Conditional formatting of an entire row based on multiple cells I'm trying to make a conditional formatting rule in Google Spreadsheet that would highlight a row green based on text in multiple cells.   If cells L2, M2, and N2 = APPROVED; and cell O2 = X; then row 2 = highlighted green. In other words, why don't we always use Schottky diodes if they are so much better? What diode properties do Schottky diodes have that makes them unfit for certain applications? I had Windows 7 and Ubuntu installed on the same partition   Some time later, decided to uninstall Ubuntu.So I booted Ubuntu,Changed the hd related to it to NTFS,I did NOT extended my main drive   restarted-> GRUB rescue&gt;. No surprise since I had just wiped out the whole GRUB directory staff   Now I just want to boot into my windows.Move my necessary files to an external memory and reinstall the windows seven     What I have in my hands: Ubuntu live CD, A copy of win7 DVD I tried to make use of the win7 DVD(I had lost my DVD) to fix the mbr but the problem is when I choose repair, It says that this version of Windows is incompatible with yours What should I do? UV Maps not showing in render I can not add all the images because the account is new and I don't have enough reputation.  I have an object (a tube of a cream) with three diferent materials, the base "plastic" material and I have created two separated materials to add the front and black labels to the tube.  The front and back label materials is the same as the "base material" but I'm trying to overlay the PNG image on top of it.  For doing that I have unwraped the parts of the object where the labels should go and placed the islands on top of the image.  I have created separated UV Maps for that purpose.     On the viewport view everything seems ok: As you can see the texture is correctly placed.   Render / Rendered view unexpected output: When rendering or in the rendered mode of the viewport the material with the UV Map appears as black.   Material nodes: For further info in case it's needed here are the screencaps of the material of the front label:        How can I solve this? Thank you! Trouble With Persistence After Making a USB Live Thumb Drive Using Pendrivelinux I created a bootable Ubuntu on my 32GB thumb drive (on a NTFS partition) and I selected a persistence of 4GB.  I've noticed that when I reboot from the initial booting off of the USB thumb drive, nothing is preserved.  No apps that were installed nor saved bookmarks.  Is there a way for me to manually force persistence? Can't learn the third Word of Power from Wulfgar Motivation behind standard deviation? Let's take the numbers 0-10.  Their mean is 5, and the individual deviations from 5 are -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5 And so the average (magnitude of) deviation from the mean is $30/11 \approx 2.72$.  However, this is not the standard deviation.  The standard deviation is $\sqrt{10} \approx 3.16$.  The first mean-deviation is a simpler and by far more intuitive definition of the "standard-deviation", so I'm sure it's the first definition statisticians worked with.  However, for some reason they decided to adopt the second definition instead.  What is the reasoning behind that decision? I would like to have an app include a custom font for rendering text, load it, and then use it with standard UIKit elements like UILabel. Is this possible? Fermat: Last two digits of $7^{355}$ I am doing this problem mentioned above and I know the answer because I know Euler's Theorem that $$a^{\varphi{(m)}}\equiv{1}\pmod{m}.$$  I used 100 as my modulus and got that the last two digits of $7^{355}$ are $43$.  My question is this.  What prime should I pick to solve this problem using Fermat's theorem $$a^{p-1}\equiv{1}\pmod{p}$$  As a precursor problem, the author (Dudley; Elementary Number Theory) asked to find the last digit of the number above; so using a modulus $5$, $7^4\equiv{1}\pmod{5}$ by Fermat.  Since the answer is $7^{355}\equiv{3}\pmod{5}$, the last digit is either $3$ or $8$.  Since $7$ is odd, the answer will clearly be odd, so $3$ is it.  The book is working linearly, building from the previous chapters and Euler's Theorem isn't mentioned for another two chapters (I remembered Euler from a problem solving class in college).    So again, what prime would be best?  How should I procede?    EDIT: I've seen a bunch of problems on the site and none that I looked at did it using Fermat.  If there is one, that could be posted and I would be happy.  EDIT 2:  it seems that the problem of finding of finding the last two digits using is rather difficult because of the implication that we need something mod 100.  Is there a way around this , like adding a step or complexifying, in order to finish the problem with Fermat? Steward badge can now be awarded multiple times -- backfill details As we announced in our post on  to the Review Queues, the Steward badge can now be earned more than once per queue. The purpose of this post is not to debate the merits of this decision ( already exists), but to explain the nuts and bolts of how it will be backfilled. This is how it will appear on your profile if you have earned it multiple times:  Once we push the visual design changes live across the network, this is how badge attainment will look on /review/close/stats:  For those of you who have already completed enough reviews to earn multiple Steward badges, we will be retroactively awarding them to you, starting today. The backfill script will run about twice an hour, and your achievements trophy will light up as each one is awarded. For those of you who have done thousands and thousands of reviews, things may be a bit noisy, so we thank you in advance for your patience. It may take up to two days for all of your badges to show up. $ \int_{0}^{ \infty} \int_{0}^{ \infty} \frac { e^{-(x+y)}}{x+y} dx dy $ 2D Integration With Quadratic Arg. of Delta Function Can I simplify an expression into form which uses my own definitions? If $\sum_{n=1}^{\infty} a_n^{2}$ converges, then so does $\sum_{n=1}^{\infty} \frac {a_n}{n}$ Let $a_1,a_2,a_3,\ldots$ be reals. Prove that if $\sum_{n=1}^{\infty} a_n^{2}$ converges, then so does $\sum_{n=1}^{\infty} \frac {a_n}{n}$.   For this I have shown the case for when $ a_n^{2} \le\frac {|a_n|}{n}$ $\Rightarrow$ $ |a_n|\le\frac {1}{n}$ $\Rightarrow$ $\frac {|a_n|}{n} \le \frac{1}{n^{2}}$ and we know that $\sum_{n=1}^{\infty} \frac {1}{n^{2}}$ converges and hence $\sum_{n=1}^{\infty}\frac {a_n}{n}$ converges by the comparison test. Now considering $ a_n^{2} \ge\frac {|a_n|}{n}$ $\Rightarrow$ $\frac {|a_n|}{n} \le a_n^{2}$ $\rightarrow$ combining the two cases for any n we have: $\frac {|a_n|}{n}\le\frac{1}{n^{2}}+a_n^{2}$ Hence using the comparion test again we know that $\sum_{n=1}^{\infty} a_n^{2}$ converges and $\sum_{n=1}^{\infty} \frac {1}{n^{2}}$ converges hence the sum converges so we can conclude that $\sum_{n=1}^{\infty} \frac {a_n}{n}$ is absoluetly convergent $\Rightarrow$ convergent.  Not to sure if this is correct, any help would be much appreciated, many thanks. Can I leave the Schengen Area (and come back) while my non-eea spouse resident card application is being processed? My wife and I recently moved to Germany. She's a Swedish citizen, and I'm American. It'll be December before I can actually apply for my residence card, because my appointment to register our address is so far out - the local police station was all booked out and we didn't get an apartment right away.  I imagine that I won't be getting my residence card until well after Christmas, by which time my 90/180 days will be passed. What if I want to leave to, say, London for the holidays? Will I be allowed back in? I hit day 90 around the 21st of December. I could maybe leave on day 89 and come back on day 90 (with my wife and our marriage cert), or use this Polish solution: , since I can't be expelled from Germany for overstaying my visa while my application is processing.  Both of those seem a bit risky though. Am I reading too far into it, or is this a legitimate concern? Will I get some sort of documentation saying I have a residence card application processing?