An instant way to resolve this $12^{3434322319}\mod7$? I tried $$12\mod7:5\mod7 \\ 12^2\mod7: 4\mod7\\ 12^3\mod7: 6\mod7\\ 12^4\mod7: 2\mod7\\ 12^5\mod7: 3\mod7\\ 12^6\mod7: 1\mod7 \\then \ the \ cycle \ repeats\\\\ 12^7\mod7: 5\mod7\\ 12^8\mod7: 4\mod7\\ ...$$ is a cycle of 6, so I'm going to calculate $$3434322319\mod6=1\mod6$$ so $12^1\mod7 = 5\mod7$ But I know there's the quickest way to resolve this, maybe using Fermat's little theorem $$a^p\equiv a \ \mod p$$ in fact $12\mod7 = 5\mod7$ but how do I know if I can apply this? I mean how can I know that $3434322319=p$ in order to apply the theorem?
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.
Is it necessary that an International cricket ground should has the boundary line in circle shape?Is it allowed by ICC to make it a square or ellipse or any other? What is the minimum and maximum area allowed for an international cricket ground? What is the maximum number of international cricket grounds that a country allowed to have? When is it appropriate to end a sentence in a preposition? Help finding this set Lets define the following:  Let A be a set. A is innumerable if and only if there exists a bijective function from A to $\mathbb{N}$  Proof that there exists an innumerable set $B \subseteq \mathcal P \left({\mathbb{N}}\right)$ such that it has the following properties:  1) $X\cap Y =\emptyset$ for all $X,Y \in B$ such that $X \neq Y$  2) $\mathbb{N}= \bigcup B$  3) All elements of B are innumerable.  Any help is appreciated. I don´t know how to find this set ... In the double-slit experiment of electrons (observed by photons), is it correct to say the collapse is caused by the momentum of the photons? I'm working off the article, . I think it's well-written, but I'm not convinced about this part:     So what is causing the change in the particle’s behaviour?      In the example given above, when electrons are fired at a fluorescent screen, when we bombard the electrons with photons the interaction changes the state of the system catastrophically as, despite lacking mass, photons do carry momentum.   Here the author strongly implicates the momentum of photons as being responsible for the collapse. But is that correct to say?  I have read that the  verified that obscuring which-way information restores wave-like behavior. So it feels to me it's more about some "outer" state (perhaps well-characterized as "information") that we don't yet understand that collapses the system, not per se the photon-electron interaction.  Or am I wrong, and in fact for example, the process of obscuring in the quantum eraser experiments somehow "undoes" the photon-electron interaction or restores the quantum state through another interaction? How can I benchmark my HDD? I've seen commands to benchmark one's HDD such as this using dd:  $ time sh -c "dd if=/dev/zero of=ddfile bs=8k count=250000 &amp;&amp; sync"   Are there better methods to do so than this? How is it that questions posted to the Stack Overflow show up as #1 in Google search results just minutes after being submitted? What SEO practices are being used to generate this sort of up-to-the-minute accuracy? Elementary central binomial coefficient estimates User is able to edit a file using vim I have a file named file.txt with some contents located in /home/myuser directory and I change the permissions of that file to root.root and 644.   -rw-r--r-- 1 root root 1.4K Jan 15 07:09 file.txt   Then I switched to myuser and opened it with vim(without sudo) and tried to edit and save it :wq! and I got output like:  "file.txt" E949: File changed while writing Press ENTER or type command to continue   Then I hit enter and then again :wq! and the output was :  WARNING: The file has been changed since reading it!!! Do you really want to write to it (y/n)?   I hit y, the file got saved and exit from vim.  The question is why a file with ROOT privileges and read-only access for a user changed by the way?  PS: 1. myuser is a sudo user.   The user permissions after saving the file changed to myuser.myuser as well.  -rw-r--r-- 1 myuser myuser 1.4K Jun 27 18:57 file.txt What is the correct formula for gravitational time dilation for a satellite in a circular orbit? How can I play the same world between PS4 and Mobile? Is there any way to get the number of pending chat request on omni channel? I have a requirement to get the number of pending chat request in the omni channel.  I have set the presence configuration capacity to 5 so that an agent can have only 5 chat requests. if a case is solved agent will get one of the pending chat request. But right now i can't see the number of pending request.     Lets say i have 10 pending request in the queue, but i don't have any numbers to see how many are pending in the queue.   Is there any way that can show the pending chat requests in the queue? I need to print an image at a certain size. What dimensions and resolution should I use? (This question is meant to be a resource to direct others to given that this is a common question.)  Given the task of creating an image that will be printed at a certain size, what pixel dimensions and resolution should I be setting my file at? One time pad: why is it useless in practice? The symmetric cryptosystem one-time pad (OTP) seems to be very beautiful since it is perfectly secret according to Shannon. Many books, however, point out the main drawback: one must create a secret key the same length as the plaintext. For this reason, the concept of perfect secrecy has these days been abandoned, and we prefer other cryptosystems.   Now, for example, suppose I want to encode my text message with the ASCII scheme, and moreover suppose that this plaintext is 10 gigabytes long, therefore (using an OTP) the key must be the same length, 10 gigabytes. I think, however, this is a ''sustainable price'' because I'm sure that the ciphertext can't be attacked.  Practically, I don't understand why having such long keys with an OTP is a big disadvantage if we reach "the dream"-- namely, perfect secrecy. Nowadays the storage of information is very easy, so what is the real problem? I can, for example, share the "long key" in person. How many connected graphs over V vertices and E edges? Is there a way to calculate the number of simple connected graphs possible over given edges and vertices?   Eg: 3 vertices and 2 edges will have 3 connected graphs But 3 vertices and 3 edges will have 1 connected graph  Then 4 edges and 3 will have 4 connected graphs  Till such values...it is easy to see its  V choose E  But what about when the number of vertices are less than number of edges...how to calculate then?  I am not able to visualize that  Can it be a variation of the Stars and Bars problem  Like...number of ways 7 edges(balls) can be connected to (kept in) 5 vertices(bags) such that no vertex(bag) is isolated (is empty)  Here maybe we might have to consider the number of edges twice as each edge needs two vertices... Total number of solutions of an equation What is the total number of solutions of an equation of the form $x_1 + x_2 + \cdots + x_r = m$ such that $1 \le x_1 &lt; x_2 &lt; \cdots &lt; x_r &lt; N$ where $N$ is some natural number and $x_1, x_2, \cdots, x_r, m$ are integers?  Also, what would be the solution in the relaxed case where $x_1, x_2, x_3, \dots, x_r$ could be equal? How can a file be copied? How do I copy a file in Python?  I couldn't find anything under . How to randomly select an item from a list? Assume I have the following list:  foo = ['a', 'b', 'c', 'd', 'e']   What is the simplest way to retrieve an item at random from this list? Error - SqlDateTime overflow. Must be between 1/1/1753 12:00:00 AM and 12/31/9999 11:59:59 PM I've been using this piece of code I've written and it's working in this most unclear manner. I wish to insert a row into the database which includes two columns of DateTime:  myrow.ApprovalDate = DateTime.Now myrow.ProposedDate = DateTime.Now   And yet, when I update the database I receive this error:     SqlDateTime overflow. Must be between 1/1/1753 12:00:00 AM and 12/31/9999 11:59:59 PM.    I've even tried copying an inserted value from the database and hard code it into the object being updated:  // I copied this value from the DB myrow.ApprovalDate =  Convert.ToDateTime("2008-12-24 00:00:00.000");   Still same error, the strange part is that the above trick worked for the first insert to the DB but failed from there on. Any ideas what's going on? What's the best solution to check a users country? I've often wondered why Stack Exchange has opted to keep the upvote/downvote buttons available for clicking on question/answers posted by the user. For instance:    Even though these buttons have no use for the user, why are they still placed on the post's left hand column? Moreover, the comments have no controls for flagging/+1'ing the user's own comment. What is the reason for this discrepancy? What benefit do the +1/-1 buttons provide to the user, even though they can't be used for anything?