For every $a \in Z$, consider the function $N:\mathbb{Z} \rightarrow \mathbb{Z}$: $$N(x)=x(2-ax)$$ Prove for every $k \in \mathbb{N}$ that $$ax \equiv1 \textrm{ mod } 10^k \implies aN(x) \equiv1 \textrm{ mod } 10^{2k}$$  Thoughts so far: We can make the following computation and find that $$aN(x)-1 = ax(2-ax)-1=-(ax-1)^2$$ which shows that $aN(x) \equiv1 \textrm{ mod } 10^k$, right? But where does $10^{2k}$ enter the picture?
I just need to check the reasoning in my proof is correct, I think it is valid although I'm not totally convinced because I can't follow the logic; does proving that $x$ is an integer prove that $ac|bd$?  Theorem: Let $a$, $b$, $c$, $d$ be integers. If $a|b$ and $c|d$, then $ac|bd$.  Proof: $aj=b$ and $ck=d$ for integers $j$, $k$. Then, $ac|bd$ implies $acx=ajck$ and thus $x=jk$ for some $x$. Since the product $jk$ is an integer, $x$ is an integer and thus $acx=ajck$ and thus $ac|bd$.
Multiple runnable classes inside JAR, how to run them? I opened terminal windows and I got a message   your screen size %d%d is bogus. Expect trouble   I closed the terminal and the next time I opened terminal, the message didn't show up.   What is this message and why did I receive it? What is `^M` and how do I get rid of it? Chat is an offshoot of the main site, and using it is  during their time here.  Chat is our place for:        Real-time collaboration   Meeting fellow members of the community   Less structured, casual (but still roughly on-topic) conversation      Our chat system exists to serve the main site; users of each site can gather and commune with each other.  When in chat, we expect all users to abide by our  policy, which means that we expect community members to treat each other with respect.    While it is a place where the community can go to interact on a more casual level, chat still needs to be moderated by the community and there are tools in place to help with the moderation. This raises a few questions, that are addressed here: Linear regression with non-symmetric cost function? I want to predict some value $Y(x)$ and I am trying to get some prediction $\hat Y(x)$ that optimizes between being as low as possible, but still being larger than $Y(x)$. In other words: $$\text{cost}\left\{ Y(x) \gtrsim \hat Y(x) \right\}  &gt;&gt; \text{cost}\left\{ \hat Y(x) \gtrsim Y(x) \right\}  $$  I think a simple linear regression should do totally fine. So I somewhat know how to implement this manually, but I guess I'm not the first one with this kind of problem. Are there any packages/libraries (preferably python) out there doing what I want to do? What's the keyword I need to look for?  What if I knew a function $Y_0(x) &gt; 0$ where $Y(x) &gt; Y_0(x)$. What's the best way to implement these restrictions? Adding script conditionally Blender 2.78 with Windows 10 lags when interacting with the UI Prove: $\displaystyle\sum_{k=1}^n k k!=(n+1)!-1$ (preferably combinatorially)   It's pretty easy to think of a story for the RHS: arrange $n+1$ people in a row and remove the the option of everyone arranged to height from shortest to highest, but it doesn't hold up for the LHS.   Alternatively, trying to visualize the LHS, I noticed that it's like a right angle tetrahedra:  1  2!+2!  3!+3!+3!  ...  But it doesn't help to see a connection to the RHS.  Note: no integrals or gamma function nor use of other identities without proving them nor generating functions. How do I log all input and output in a terminal session? How do I keep logs of all the work done via terminal? For example, I want a record of all the work done on database via terminal. Symmetric Matrix , Eigenvectors are not orthogonal to the same eigenvalue. I know that the symmetric matrix has orthogonal eigenvectors corresponding eigenvalues. Also, the eigenvectors from the same eigenvalue are linearly independent.  I need an example of symmetric matrix such that the eigenvectors from the same eigenvalue are not orthogonal.  Also, I know that the eigenvalues of the symmetric matrix are real. But , this statement true iff the entries are real numbers.Right? Since the matrix 2x2 $$A=\begin{bmatrix} 1 &amp;i \\  i &amp;1  \end{bmatrix}$$  has non real eigenvalues When I slam my hand onto my keyboard, only a maximum of 6 keys will be registered, I've tested this for the past few minutes and was unable to get more than 6. Is this the same on all keyboards? If not, what would be the distinguishing factor, If yes: why 6 specifically?  For clarification: I do not have a usecase for this, I am simply curious Do we know how Senju Hashirama died? If 20 independent Bernoulli trials are carried out each with a different probability of success and therefore failure. What is the probability that exactly n of the 20 trials was successful?  Is there a better way of calculating these probabilities rather than simply summing together the combinations of success and failure probabilities? What points in the complex plane satisfy $|z|=arg(z)$. Why does:   $$ \sum_{k=0}^{n} k \binom nk p^k (1-p)^{n-k} = np $$ ? Taking the derivative of:   $$ \sum_{k=0}^{n}  \binom nk p^k (1-p)^{n-k} = (1 + [1-P])^n = 1 $$   does not seem useful, since you would get zero. And induction hasn't yet worked for me, since – during the inductive step – I am unable to prove that:   $$ \sum_{k=0}^{n+1} k \binom {n+1}{k} p^k (1-p)^{n+1-k} = (n+1)p  $$   assuming that:   $$ \sum_{k=0}^{n} k \binom {n}{k} p^k (1-p)^{n-k} = np  $$ Making a Laser in Cycles - Straight Beam of Light? How to exclude a package from apt-get autoremove? I'm in a situation where the list of orphaned packages to be removed by apt-get autoremove includes a package that I want to keep. Apparently I have accidentally removed a package which was depending on it. How can I now mark the package as explicitly desired, so that apt-get autoremove will not remove it? Python and regular expression with Unicode I need to delete some Unicode symbols from the string 'بِسْمِ اللَّهِ الرَّحْمَٰنِ الرَّحِيمِ'  I know they exist here for sure. I tried:  re.sub('([\u064B-\u0652\u06D4\u0670\u0674\u06D5-\u06ED]+)', '', 'بِسْمِ اللَّهِ الرَّحْمَٰنِ الرَّحِيمِ')   but it doesn't work. String stays the same. What am I doing wrong? Why does DMM show AC and DC current when source is a battery? (How to calculate power in this case?) The circuit is an altered Joule Thief of my design that is driven by an AA or AAA battery. I’m observing the current at the battery anode with my DMM in series. (Extech EX330) This circuit is extremely efficient (In relation to other Joule Thief I've build.) and does some weird stuff - like, it needs a pushbutton sometimes.    Measurements: It is drawing 9.21 mA DC and 6.48 mA AC in series with the battery. It shows 1.364 VDC and 0.010 VAC in parallel with the battery. Output on the LED is 2.572 VDC and 0.011 VAC with no measurable frequency. (with my DMM at least) All measurements are taken with the same configuration and voltage.    But how is it drawing AC and DC and how do I calculate power consumption? --> Could it be because the emitter pin of the NPN is directly connected to ground? --> Or is the circuit just "switching on and off"?     Download schematic:  (I tried to simulate the circuit in LTspice. Thanks for making me aware of this amazing program, Sredni Vashtar!)     What the pushbutton is used for: The pushbutton is used to "kickstart" the oscillation when it wont start by itself. For example: I have to manually "oscillate" using this button to start the circuit if the capacitor is too big and battery voltage is low. Pushing the button releases the energy in the transformer into the capacitor. If it reaches LED forward voltage the circuit keeps oscillating on it's own. It does work without manual intervention when then capacitor in parallel with the LED is not too big, like &lt;5000 µF, and it's a fresh battery.    . Thank you for your answers! I'm learning so much! :D What does "Package  has no installation candidate" mean? What does the above Error mean? What is an installation candidate?  I was trying to do  sudo apt-get install munin-memcached     and I get this message:  Reading package lists... Done Building dependency tree        Reading state information... Done Package munin-memcached is not available, but is referred to by another package. This may mean that the package is missing, has been obsoleted, or is only available from another source E: Package munin-memcached has no installation candidate   I googled for the problem and someone said to do a apt-get upgrade but it still did not solve my problem.