Calculate $6^{1866}$ in $\mathbb{Z}_{23}$  $Solution:$ Note that $1866=22\cdot 84 + 18$ then by Fermat's theorem  $$[6^{1866}]=[6^{22}]^{84}[6^{18}]=[1]^{84}[6^{18}]=[6^{18}]$$ Then $6^6=46656=2028\cdot 23 + 12$ It is true that $$[6^{1866}]=[6^6]^3=[-11]^3=[121][-11]=[72]=[3]$$  So $ 6 ^ {1866} $ is $3 $ in $ \mathbb {Z} _ {23} $, is that correct? Thank you for reading.
I've read about Fermat's little theorem and generally how congruence works. But I can't figure out how to work out these two:   $13^{100} \bmod 7$ $7^{100} \bmod 13$   I've also heard of this formula:  $$a \equiv b\pmod n \Rightarrow a^k \equiv b^k \pmod n $$  But I don't see how exactly to use that here, because from $13^1 \bmod 7$ I get 6, and $13^2 \bmod 7$ is 1. I'm unclear as to which one to raise to the kth power here (I'm assuming k = 100?)  Any hints or pointers in the right direction would be great.
A Regular Tetrahedron is a cool Polyhedron. A regular tetrahedron has this property:  For any two of its vertices exists a third vertex, which forms a regular triangle with these 2 vertices.(It doesn't necessarily have to be a face of it).  Are there any other polyhedrons that have the same property?  I think there isn't such a polyhedron. But not sure how to prove it. I've tried proving that there isn't an irregular tetrahedron with this property, assuming that there is a pair of unequal sides.   Current answers are a little confused with the conditions of the problem. How is Gauss' Law (integral form) arrived at from Coulomb's Law, and how is the differential form arrived at from that? I've always wondered this - why can't you declare variables after a case label in a switch statement?  In C++ you can declare variables pretty much anywhere (and declaring them close to first use is obviously a good thing) but the following still won't work:  switch (val)   {   case VAL:     // This won't work   int newVal = 42;     break; case ANOTHER_VAL:     ...   break; }     The above gives me the following error (MSC):     initialization of 'newVal' is skipped by 'case' label   This seems to be a limitation in other languages too.  Why is this such a problem? This is probably pretty basic, but to save me an hour or so of grief can anyone tell me how you can work out the number of bits required to represent a given positive integer in Java?  e.g. I get a decimal 11, (1011). I need to get the answer, 4.  I figured if I could work out how to set all the bits other than the most significant bit to 0, and then >>> it, I'd get my answer. But... I can't. Why comment reply auto-completion feature (@) sometimes doesn't work? Why didn't the Ministry use Veritaserum to question criminals? The truth potion (Veritaserum) is one of the most powerful potions that forces the drinker to speak truth. Then why didn't Ministry used it in any criminal cases instead of long discussions and proceedings? I know there are antidotes to the potion (Dumbledore tells Harry in the 6th book , "Slughorn might have been carrying antidote to veritaserum ever since I asked him about the memory"), but why don't keep criminals in custody until the effect of the antidote wears out and then make them drink veritaserum. Won't that result in perfect judgement? Singular or plural verb after a series connected by "or" The following sentence refers to an apocalyptic story where money no longer has any value:     A can of sardines, radio batteries, or a bicycle is/are more precious than money.   Should I use is or are after the series? If $F\supset E$ and $E\supset k$ are normal extensions. I want a counter-example where $F\supset k$ is not normal How to create a Wi-Fi Hotspot (in Access Point mode)? 2 orderby in wp_query with 2 custom fields I'm triyng to display hotels (taxonomy which i have created) by two 'orderby' with wp_query but i'm not success.  Results looks likes:  Hotel 1 :       2 persons          5 stars Hotel 2 :       2 persons          4 stars Hotel 3 :       2 persons          3 stars Hotel 4 :       3 persons          5 stars Hotel 5 :       5 persons          5 stars Hotel 6:        10 persons         2 stars Hotel 7:        10 persons         1 star   The first 'order by' is on the number of persons(the biggest in first) and the second 'order by' is on the rating in stars (so, if several hotels have the same number of persons, the hotel with the biggest stars rating would be in first).  Below my code. I have tried to add an array inside an array with meta_query but it doesn't work.  $args=             array(                                     'post_type' =&gt; 'hotel',                                     'Services' =&gt; $Services,                                     'Town' =&gt; $Town,                                      'meta_key' =&gt; 'Number_persons',                                     'orderby' =&gt; 'meta_value_num',                                     'order' =&gt; 'ASC',                                        'meta_query' =&gt; array(                                                            array(                                                                'meta_key' =&gt; 'Rating',                                                                'orderby' =&gt; 'meta_value',                                                                'order' =&gt; 'DESC',                                                               )                                                        )                  );   $the_query = new WP_Query( $args );   Thanks a lot :) Performance of Non Clustered Indexes on Heaps vs Clustered Indexes compares the performance for individual select/insert/delete/update and range select statements on a table organized as a clustered index vs that on a table organized as a heap with a non clustered index on the same key columns as the CI table.  Generally the clustered index option performed better in the tests as there is only one structure to maintain and because there is no need for bookmark lookups.  One potentially interesting case not covered by the paper would have been a comparison between a non clustered index on a heap vs a non clustered index on a clustered index. In that instance I would have expected the heap might even perform better as once at the NCI leaf level SQL Server has a RID to follow directly rather than needing to traverse the clustered index.   Is anyone aware of similar formal testing that has been carried out in this area and if so what were the results? What is the difference between the two "worlds" when going thru the day/night wormhole at the altar? Visa for European countries for US passport holder I'm a US passport holder. What documents do I need when I fly from the U.S. to Germany?  Assume I'm going for a few weeks of vacation.  Obviously I need my passport.  What about a "visa"?  Anything else? How to use a field value in Template.php Which one sounds better? "of mine" or "my" My friend (or "a friend of mine"?) told me that anytime I need to express such expression, it's always better to say "of mine", because it sounds more naturally.  Like "two brothers of mine" rather than "my two brothers".  Is that true? Let $M=\big( \begin{smallmatrix} A & B \\ C & D \end{smallmatrix} \big)$. Prove $\det(M)=\det(A)\cdot \det(D-C·A^{-1}·B)$. Let  (shown in matrix blocks) $M=\big( \begin{smallmatrix} A &amp; B \\ C &amp; D \end{smallmatrix} \big)$ a square matrix such that $A$ is invertible and $D$ is a square matrix. I have to prove that  $$\det(M)=\det(A)\cdot \det(D-C·A^{-1}·B)$$  I also have an indication to use it: Consider previously these cases: \begin{align*} A_1 &amp;= \begin{pmatrix} A &amp; 0 \\ 0 &amp; I \end{pmatrix},\\ A_2 &amp;= \begin{pmatrix} I &amp; 0 \\ 0 &amp; D \end{pmatrix},\\ A_3 &amp;= A_1·A_2,\\ A_4 &amp;= \begin{pmatrix} I &amp; A^{-1}·B \\ 0 &amp; I \end{pmatrix},\text{ and}\\ A_5 &amp;= A_3\cdot A_4. \end{align*}   The problem is difficult in general for me and I don't even know how to use the indication. In music, when two or more pitches are played together at the same time, they form a chord. If each pitch has a corresponding wave frequency (a pure, or fundamental, tone), the pitches played together make a , which is obtained by simple addition. This wave is no longer a pure sinusoidal wave.  For example, when you play a low note and a high note on a piano, the resulting sound has a wave that is the mathematical sum of the waves of each note. The same is true for light: when you shine a 500nm wavelength (green light) and a 700nm wavelength (red light) at the same spot on a white surface, the reflection will be a superposition waveform that is the sum of green and red.  My question is about our perception of these combinations. When we hear a chord on a piano, we’re able to discern the pitches that comprise that chord. We’re able to “pick out” that there are two (or three, etc) notes in the chord, and some of us who are musically inclined are even able to sing back each note, and even name it. It could be said that we’re able to decompose a Fourier Series of sound.  But it seems we cannot do this with light. When you shine green and red light together, the reflection appears to be yellow, a “pure hue” of 600nm, rather than an overlay of red and green. We can’t “pick out” the individual colors that were combined. Why is this?  Why can’t we see two hues of light in the same way we’re able to hear two pitches of sound? Is this a characteristic of human psychology? Animal physiology? Or is this due to a fundamental characteristic of electromagnetism? Is the tabu package obsolete? Consequences of a master thesis with errors I finished my master thesis 6 years ago, since then I gained a lot of experience on my field and also in academia. I can say that in that time I was a "child" , very inexperienced. I think that also my advisor did not help a lot, and I tried to do  a lot of things by my own, without someone to really review my things. Despite that, I still received a good mention by the master committee on final results.   Some days ago, I just decided to read part of master thesis, and I just realized a big nightmare coming. I identified a lot of typos, some references errors, and some improvements that I should do, but I did not. The worst thing is that I wrote a paper to a conference with my master thesis results (and some errors are there). The conference is not a high impact conference, but still the paper has my name and is available for everyone...   As I am finishing my PhD studies for now, I am concerning a lot that situation. My dream and goal is to pursue a Professor career. So, my questions are: What are (or could be) the consequences of a master thesis and paper with errors (some minor and some major) in academia? Revoke degree after years by errors in master is possible ( I am concerning a lot this situation that I thought on it)? How can I minimize it (I thought that I can put a fixed version of my thesis and paper in the future, for instance on my personal website)? How those errors can impact me in a Professor recruitment process?  Sorry for the long text, I am blaming myself all the time, and really afraid about the possibilities since I know that Academia career is really competitive. How did Sauron remain in control of the Nazgûl?