Prove that If gcd$(a,2)=1,$ and $p$ is an odd prime, then $(a^{p-1})-1$ is divisible by $8$.
I am trying to show that if $n$ is an odd natural number, then $8$ divides $n^{2}-1.$ I was able to prove that because I know that if $n$ is an odd natural number, then $n^{2}$ can be written as $8k+1$ for some $k\in \mathbb{N}.$ I would like to show this question by using the Euclidian division. Then I wrote $n^{2}-1=8k+r$, where $0\leq r  &lt; 8.$ When $r=2$ we get $n^{2}-1=8k+2.$ Since $n$ is odd, then $n^{2}-1$ is even and I got stuck. Is there way to fix that?
In a moment of cheapness, I decided not to order a 5v to 3.3v level shifter from Sparkfun, but instead to put it together myself.  The  used a  MOSFET, but since I enjoy surface-mount soldering about as much as root canal treatment, I decided to use a  that is through-hole mounted and easily available from my preferred supplier.  The results of this were sub-optimal.  When pulling the 5v line to ground, all was fine - the 3.3v side went to 0.07v.  But when pulling the 3.3v to ground, the 5v line showed around 4.14v (it was otherwise a touch above 5.1v).  On a closer reading of the datasheets involved, as well as the original Philips application note on the subject, I began to come to the conclusion that the gate threshold voltage was the problem.  By altering the schematic and tying the MOSFET's gate to 5v instead of 3.3v, both sides seem to work fine.  Pulling either side low causes the other side to go low.  I'm not, however, at all confident that this is actually a sane thing to do.  My understanding of the original schematic isn't deep enough to form a sensible opinion.  Will this modified schematic work, or are the current good results I'm seeing mere fluke or a precursor to something releasing magic smoke? How would I import/export Blender's preferences? Series about proxy space battles via implanted computers in kids The premise of the story is that there are 2-3 main coalitions on Earth and they fight over resources in the edges of the solar system with spaceships that are controlled from Earth by kids with computers in their heads.  If I recall correctly:   The main character's name is Tom, he starts out by conning people at some sort of gambling arcade before he is found and recruited by the "army" The Pentagon in the US has a super tall tower built in the courtyard called The Spire where most of the three books take place. The reason they need kids for this has something to do with brain elasticity and putting computers in adults causes them to go mad. All the cadets are super dismissive about learning how to program their computer aside from one girl who is super good at it. The best fighters from each side of this conflict were something of superstars, but everyone else was kept completely anonymous for their protection. There were at least 3 books last I knew, released circa 2011-2014 The third book ends with Tom being the first astronaut to Mars. The first book had a green cover, the second was yellow, and the third was red. Though I might have the order wrong. How can I detect when an Exception's been thrown globally in Java? How can I detect when an Exception has been thrown anywhere in my application?  I'm try to auto-magically send myself an email whenever an exception is thrown anywhere in my Java Desktop Application.  I figure this way I can be more proactive.  I know I could just explicitly log and notify myself whenever an exception occurs, but I'd have to do it everywhere and I might(more likely will) miss a couple.  Any suggestions? Is $\sqrt{64}$ considered $8$? or is it $8,-8$? how much money it takes for a day without including accommodation In Uk How to stop iPhone users from sending iMessages and send SMS instead I used to have an iPhone, but I changed to an android (shock, horror!). Unfortunately Apple refuses to let me go and whenever friends message me the messages get sent as iMessages (unless they remember to specifically chnge the format). I never get these messages. They never get resent as SMSs automatically as is supposed to happen. How can I get my phone number deleted from the iMessage list. I've turned off iMessage on the old iPhone. Should I really pay off my entire credit card balance each month or should I maintain some balance? Bound states of the $V(x)=\pm \delta'^{(n)}(x)$ potential? The $\delta(x)$ Dirac delta is not the only "point-supported" potential that we can integrate; in principle all their derivatives $\delta', \delta'', ...$ exist also, do they?  If yes, can we look for bound states in any of these $\delta'^{(n)}(x)$ potentials? Are there explicit formulae for them (and for the scattering states)?  To be more precise, I am asking for explicit solutions of the 1D Schroedinger equation with point potential,  $$- {\hbar^2 \over 2m} \Psi_n''(x) + a \  \delta'^{(n)}(x) \Psi(x) \ = E_n \Psi_n(x) $$  I should add that I have read at least of three set of boundary conditions that are said to be particular solutions:   $\Psi'(0^+)-\Psi'(0^-)= A \Psi(0)$ with $\Psi(0)$ continuous, is the zero-th derivative case, the "delta potential". $\Psi(0^+)-\Psi(0^-)= B \Psi'(0)$ with $\Psi'(0)$ continuous, was called "the delta prime potential" by Holden. $\lambda \Psi'(0^+)=\Psi'(0^-)$ and $\Psi(0^+)=\lambda\Psi(0^-)$ simultaneusly, was called "the delta prime potential" by Kurasov   The zero-th derivative case, $V(x)=a \delta(x)$ is a typical textbook example, pretty nice because it only has a bound state, for negative $a$, and it acts as a kind of barrier for positive $a$. So it is interesting to ask for other values of $n$ and of course for the general case and if it offers more bound states or other properties. Is it even possible to consider $n$ beyond the first derivative?  Related questions  (If you recall a related question, feel free to suggest it in the comments, or even to edit directly if you have privileges for it)  For the delta prime, including velocity-dependent potentials, the question has been asked in   In the halfline $r&gt;0$, the delta is called "Fermi Pseudopotential". As of today I can not see questions about it, but  seems to be the same potential.  A general way of taking with boundaring conditions is via the theory of self-adjoint extensions of hermitian operators. This case is not very different of the "particle in 1D box", question  A general take was the question  A related but very exotic question is  because obviosly the point-supported interacctions have a peculiar scaling  Comments  Of course upgrading distributions to look as operators in $L^2$ is delicate, and it goes worse for derivatives of distributions when you consider its evaluation $&lt;\phi | \rho(x) \psi&gt;$. Consider the case $\rho(x) = \delta'(x) = \delta(x) {d\over dx}$. Should the derivative apply to $\psi$ only, or to the product $\phi^*\psi$? What is the word that means “that which can be fixed up at home”?   For example, I buy parts from a store, bring the parts home, and then can construct the thing out of those parts by myself. What is the word that means such an object, one that can be made at home in a do-it-yourself style? Is $\emptyset \in \emptyset$ or $\emptyset \subseteq \emptyset$? filter the file type with the file upload control Two trains travel on the same track towards each other, each going at a speed of 50 kph. They start out 100km apart. A fly starts at the front of one train and flies at 75 kph to the front of the other; when it gets there, it turns around and flies back towards the first. It continues flying back and forth till the two trains meet and it gets squashed. How far did the fly travel before it got squashed?  The question is relatively simple - the trains take an hour to collide, and in an hour the fly would travel 75 km. However, the question got me wondering: Lets say you are given a time $t$ in hours(where $t&lt; 1$) How many &quot;rounds&quot; would the fly make in $t$ time? A round  is defined as moving from one train to another and back. How would I model the equation? Is it possible to reintegrate a Horcrux? Will I be demoted to a lower league in Starcraft 2 if I don't play? I've just been promoted from platinum league to diamond and master pretty fast (in 4 vs 4). This was totally amazing for me but I'm afraid I will be demoted again to a lower league as fast as I came higher.   If I don't play any 4 vs 4 anymore, will they demote me to a lower league automatically or do they only do so, if I play? permission denied BASH How can I restart an app game from the beginning I took an English assessment test online and this was my answer:     Someone suggested to go for a walk.   My answer was wrong and this was the correct sentence:     Someone suggested going for a walk.   Can someone please explain why my answer was wrong? How can I horizontally center a &lt;div&gt; within another &lt;div&gt; using CSS? &lt;div id=&quot;outer&quot;&gt;   &lt;div id=&quot;inner&quot;&gt;Foo foo&lt;/div&gt; &lt;/div&gt; How come Snape and Lily have the same patronus? Why is Severus Snape's Patronus the same as Lily Potter's? Are Snape and Lily related somehow? Are Snape and Harry related?