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I came across a combinatorial proof of the Fermat's Little Theorem which states that    If $p$ is a prime number then the number ($a$$p$-$a$) is a multiple of $p$ for any natural number $a$.  Let me write down the proof.    PROOF  We have pearls of $a$ colours . From these we make necklaces of exactly $p$ pearls .   First,we make a string of pearls . There are $a$$p$ different strings.If we throw away  the $a$ one one -coloured pearls  ($a$$p$-$a$) strings will remain.We connect the ends of each string to get necklaces.We find that two strings that differ only by a cyclic permutation of its pearls result in indistinguishable necklaces.But there are $p$ cyclic permutations of $p$ pearls on a string . Hence the number of distinct necklaces is    [($a$$p$-$a$)/$p$].  Because of this interpretation this is an integer.Thus ($a$$p$-$a$) is a multiple of $p$ for any natural number $a$. [HENCE PROVED]     MY CONFUSIONS    What is the use of the fact that $p$ is a prime in this proof?  If this proof is true then why is this not true for every positve integer $p$? I got the Turkey e-visa several times with my schengen visa. In the e-visa form there is a condition      I have a round-trip ticket with Turkish Airlines, Pegasus Airlines,   Onur Air or Egypt Air.   What if I am traveling from another country to Turkey or using another airlines like Nile Air form Egypt? I'm more of a math guy than a physics guy so bear with me....  In fractal geometry, fractals are considered to have fractional dimension. For instance an object such as the Koch curve has a fractal dimension of around $1.2$. You can also define a concept called fractal extent (which is the experimental version of hausdorff measure). This extent is a measure of how large a fractal is. It's even been used to more accurately model the distribution of coastline bird nesting.   Even though fractals work as an interpretation of geometrical units, I don't think they would describe things such as $kg^{1/2}$ very well. In addition, I've heard about white Gaussian noise that has square root time units.   What is the physical meaning of fractional units such as the above? (Since it obviously has little geometric meaning) Formal Proof that area of a rectangle is $ab$ I tried to prove that the area of a rectangle is $ab$ given side lengths $a$ and $b$.  The best I can do is the assume the area of a $1\times1$ square is $1$. Then not the number of $1\times1$ squares that fit in an $a\times b$ rectangle is $ab$. Therefore area is $a\cdot b$. This does not seem rigorous however. How to run two Thread Group in a loop - JMeter I'm facing problem here, There are 2 Thread Groups  and both are correlated ( run thread group consecutively is ticked in test plan ).  1st is user thread group and 2nd is Driver thread group. So thread group running concurrently but it is not happening in twice. I've put loop count for 2 they will run each independently after completion of first cycle. I want run this flow such a way that after completion of driver group request next do second iteration (start from user) thread group. So how to do this stuff ?  Loop will be :   User thread group Driver Thread group   After completing this once again start with :   User thread group Driver thread group   like this it should(loop) happen for multiple times Good reasons to prohibit inheritance in Java? How to switch from Resident visa to Tourist visa status in the Schengen area? How do you fix a bad merge, and replay your good commits onto a fixed merge? How to make a standalone document with one equation? What is the motivation for including the compactness and semi-simplicity assumptions on the groups that one gauges to obtain Yang-Mills theories?  I'd think that these hypotheses lead to physically "nice" theories in some way, but I've never, even from a computational perspective. really given these assumptions much thought. Google Maps choose zoom to fit country TIGHTLY in given window How do I find the right size of filters for a lens? My last name is not the same on ticket as on passport; rest of details are same How to find all positive integers $n$ such that $(n-1)!+1$ can be written as $n^k$ for some positive integer $k$? Why would a company like Apple be buying back its own shares? What is the reason that a company like AAPL is buying back its own shares? Does it have too much cash and it doesn't know what to do with it? So it is returning the cash back to investors? If an polynomial has a complex root, is it necessary that its conjugate is also a root? If an polynomial has a complex root, is it necessary that the conjugate of that root is also a root of the polynomial? I am not sure about this. Please comment on this. I often see "replace with" and "replace by" used interchangeably, but this doesn't sound right to me:     I replaced that component by this one.   I would use "with" in such a sentence. "By" only seems reasonable in passive, although "with" sounds like it would there work too:     That component was replaced by this one.   That component was replaced with this one.   In my native language, the equivalent of "replace by" can only be used in passive, and even then it's a bit weird unless a person is the object replacing something – perhaps this affects my judgment?  Web searches haven't come up with anything conclusive; the results are contradictory and speculative at best.  Are "replace with" and "replace by" interchangeable in active context? What about passive? Are there stylistic reasons to prefer one over the other? Microphone Not Working With FaceTime Multiple logistic regression power analysis I just want to write \Sha without ruining everything I realize similar questions have been asked before but I am not satisfied with any of the answers I've seen. I would like to use the Cyrillic letter Ш to denote a particular group in mathematics. Most solutions involve using OT2 or T2A encoding; for example, at . However, this makes the other text in the document look bad.   For example, using the solution above, I get    while I would like to have     The first seems to be shaded strangely and the characters don't seem to align right.  Does anyone know how I can write the character Ш (in math mode) without changing the default font for the rest of the text?