I'm wondering the following: I know that $a \equiv b \textrm{ mod } m \implies ac \equiv bc \textrm{ mod } m$ for $a,b,c,m \in \mathbb{Z}$ and $m \neq 0$. I'm wondering that if we know that $\gcd(c,m) = 1$ and that $ac \equiv bc \textrm{ mod } m$, is $a \equiv b \textrm{ mod } m$? If it is true, how would we prove it?
How would you show that if $ac≡bc$ $\mod m$ and $\gcd(c,m)=d$, then  $a≡b$ $\mod \frac{m}{d}$?  Any help would be much appreciated!
I have some doubts regarding net run rate in the ongoing ICC world twenty 20 2014 event.It takes a while to read so please don't hesitate and give me a solution regarding this.    My doubt is that in match 12 of world twenty 20 2014(In Group Stage Last Match on march 21 2014) () Wesley Barresi of Netherlands hit a six when the team score was 187 while chasing Ireland score of 189, That take Netherlands score to 193.  If that is the case in match 17 between India and West Indies in match of super10 group (in ICC world twenty 20 2014 on march 23 2014) () while Suresh Raina (India) hit a four when the team score was 129 while chasing West Indies score of 129,That takes India Only To 130. why Only One run is added to the score in this match(match 17 on march 23 2014) while 6 runs are added to the score in match 12 on march 21 2014.  I don't understand the strategy behind this.Because if India's Score is taken as 133 then India's net run rate as of now(24 march 2014) will be 0.442(Runs Scored For 264 in 38.1 overs ) But Now it is only 0.383(Runs Scored For 261 in 38.1 overs ).As the net run rate plays an important role in the tournaments like this i am not very happy with this.  As being an Indian Fan now a days I look a lot in to net run rate as it has eliminated India in many tournaments including last ICC world twenty 20 2012.  Is There Any One who can clarify my doubts regarding the net run rate as i want to be sure of what was happening? How to hide a folder in sdcard in android It is given $23!=2585201xy38884976640000$. Now it is required to find the value of $x$ and $y$. I know I could find it by using divisibility rules and solving simultaneous equations. Is there any other way to solve it (without computing it by a calculator)? This question is just out of curiosity. How do you "OR" criteria together when using a criteria query with hibernate? Does the mass of a battery change when charged/discharged? If so, how much? Is it possible to detect it, or is it beyond any measurement?  I'd say there are two possible scenarios (depending on the battery type) and both seem interesting:   The battery reacts chemically with its environment. The battery doesn't exchange any matter with its environment except electrons. I suppose there should be some difference at least due to the principle of energy-matter equivalence, but the difference is most likely immeasurable. System freezes completely with Intel Bay Trail My system freezes completely at random, frequent intervals. I started to have the same problem in Ubuntu 14.04 but after recent upgrade to 16.04 there is no improvement, in fact it seems worse.  When it happens, it's impossible to do anything. I've tried everything in this thread:  but nothing works, I have to hard reset. I have read all the system logs and journalctl but there is never any information that could help diagnose the problem.  This is a dual-boot system with Windows 10 and there's no problem there, so it's not defective hardware.  My laptop has an Intel Bay Trail processor (Pentium N3540) Why was Edo Tensei Madara young, when he had died at an old age? From Naruto manga and anime, I know that Uchiha Madara died as an old person after he helped Uchiha Obito.   But it was bit weird for me, when Kabuto summoned Uchiha Madara-sama with Edo Tensei, Madara was a younger person.  Again, when Chiyo was summoned, she was older as when she died.  How can it be possible? Quantum Entanglement - What's the big deal? Bearing in mind I am a layman - with no background in physics - please could someone explain what the "big deal" is with quantum entanglement?  I used to think I understood it - that 2 particles, say a light-year apart spatially, could affect each other physically, instantly.  Here I would understand the "big deal".  On further reading I've come to understand (maybe incorrectly) that the spatially separated particles may not affect each other, but in knowing one's properties you can infer the other's.  If that it the case, I don't see what the big deal is... 2 things have some properties set in correlation to each other at the point of entanglement, they are separated, measured, and found to have these properties...?  What am I missing? Is it that the particles properties are in an "un-set" state, and only when measured do they get set?  (i.e. the wave-function collapses).  If this is true - why do we think this instead of the more intuitive thought that the properties were set at an earlier time? When I launch an app, the focus doesn't move to the opened app Let's say I have Gthumb open, and then launch Nautilus from the Unity launcher (in 12.04). Why does it sometimes not focus the mouse on Nautilus, which I just launched? Instead, the focus remains on Gthumb while Nautilus opens in front of Gthumb (and not being focused on).  Is this correct behaviour? Change Data Type Varchar To Varbinary(max) In SQL Server Did Sam's memories transfer to the displacee when he leaped? In the TV show Quantum Leap Sam Beckett would travel history and "right" wrongs, temporarily displacing someone native to that timeline in the process. Would the memories of what Sam did transfer to the person he displaced upon his leaping out, or would they wind up with amnesia for the time Sam was there? Intuition for Rayleigh PDF We have ground-truth data $\mathbf{x}^* = (0,0)^T \in \mathbb{R}^{2}$. Furthermore, we have $N$ measurements $\mathbf{x}_i \in \mathbb{R}^{2}, i\in \{1,\ldots,N\}$. We calculate the $N$ error vectors $\mathbf{e}_i = \mathbf{x}_i - \mathbf{x}^*$. We assume that the two components of $\mathbf{e}_i = (e_{x,i}, e_{y,i})^T $ are independent of each other and normally distributed with equal mean $\mathbf{\mu} = (0,0)^T$ and variance $\sigma^2$. The absolute errors $e_i = \sqrt {e_{x,i}^2 + {e}_{y,i}^2} $ follow a Rayleigh distribution, is that correct?  The probability density function (PDF) of the Rayleigh function is zero at a value of zero. Why is this the case (intuitively)? I would assume that our errors should most likely be 0 (i.e. PDF should be highest at 0)... I'm having a difficulty in finding one way car rentals from Alaska, which I can return in Seattle.  Do all the main rental agencies restrict such one-way rentals, as they involve crossing through Canada, before entering the lower 48 states? problems installing mathtools Conservation law of energy and Big Bang? Did the law of conservation of energy apply to the earliest moments of the Big Bang? If so, what theoretical physics supports this?   I hear that Einstein's theory of relativity disputes the law of conservation of energy- so does that mean the law is false, or only some aspect of it? I have an external sound card which mostly works fine, except that when its first plugged in, or when I turn on my laptop, the volume gets set to 100%. That is, the overall system volume, as shown in the sound indicator.  What I'm looking for is a terminal command that will set that volume to 50%, so that I can run it on login and not have to worry that the first audio I play is going to blare out at me if I forget to turn the volume down. What commands allow you to change that volume, i.e. the one in the sound indicator?  I've poked around in gsettings and dbus, but have been unable to find anything that would let me set the system volume like that. I do know about alsamixer, but that's not the solution I'm looking for, since that doesn't allow me to change the volume using the sound indicator or my laptop's multimedia keys. How can I subdivide this plane into even squares? The  spell says:     The spell captures some of the incoming energy, lessening its effect on you and storing it for your next melee attack. You have resistance to the triggering damage type until the start of your next turn. Also, the fist time you hit with a melee attack on your next turn, the target takes an extra 1d6 damage of the triggering type, and the spell ends.   The  cantrip says:     Lightning springs from your hand to deliver a shock to a creature you try to touch. Make a melee spell attack against the target. You have advantage on the attack roll if the target is wearing armor made of metal. On a hit, the target takes 1d8 lightning damage, and it can’t take reactions until the start of its next turn.   If I absorb fire damage with absorb elements, would I be able to add an extra 1d6 fire damage to my next shocking grasp attack? Map matching links and ideas? I'm using OpenStreetMap and its vectorial road network and I'd like to implement a map matcher algorithm.   Currently I'm able, for each GPS position, to retrieve the nearest road segment and calculate the projection of this position to that segment, like on this image (Red pin is the pure GPS position, in blue the mapped segment and in Green the mapped position):    However, due to the lack of accuracy of the GPS, sometimes the mapped position jumps from segment to another and can provide some inconsistent mapped position from time to time.  My current algorithm is very basic : from the pure GPS position, I get the nearest segment and decide that the mapped matched position is on this one. I know that this can be really improved.  I can imagine that taking the vehicle direction into account will improve the map matching but do you know any other approach that would enable me to improve my map matcher ?  I seek any link, and/or open source software? Does an Animagus's animal form change along with their Patronus? In the two examples shown of an Animagus capable of casting a corporeal Patronus (Minerva McGonagall and James Potter) both had the same animal as their Patronus and as their Animagus animal.  Now, it is also shown that extreme circumstances can change one's Patronus. I think the only example of this was Tonks, which turned from a jackrabbit into a wolf in response for her love of Lupin.  If Tonks (or any other wizard who had their Patronus change) had been an animagus, would her animagus have changed as well?