For any natural number $a$, prove that $a^2$ when divided by $5$ cannot have a remainder of $3$. (Hint: What are the possible values of the remainder when $a$ is divided by $5$?  Using the hint, I found that the possible values of the remainder are $1, 2, 3, 4$, and $0$, and $3$ is only a remainder when the last digit of $a$ is $3$ or $8$.  But I'm not sure how to explain this in a proof, and I don't know how to extend it to $a^2$. Any help would be much appreciated!
Show that there is no solution to the congruence $x^2\equiv3\bmod5$.   I'm not sure if the fact that the only numbers divisible by $5$ are those ending in $0$ or $5$ will help&hellip;
Using `\input` to insert a section of a document into another Is there any way one can use \input to insert only a specific section of a document into another? I intend on using this to improve the readability of an article with lots of pgfplots graphs in it, by moving all of my graphs to another file, and using something like \input{file.tex:graph1} to insert each graph where it is needed, rather than have all of them displayed together In Volo's Guide it is suggested that a curse could transform someone into a race of 'monstrous origin' (p. 118). Does this apply to the Bestow Curse spell specifically? Note: this is a  (my apologies).  In film / lore, such magic does , ,   - or even  transformative curses.  Points &amp; Concerns:   this is within guidelines for a 'reasonable' curse, i.e. less harm than the listed set Bestow Curse writeup encourages DM-homebrewing &amp;/or interpretation. Great role-play possibility / RAI (hence most myths, fairy tales, cartoons, legends, etc) this could be subject to abuse, temptation to bestow a 'curse' as a blessing / boon one could also change: age, size, weight &amp;/or gender (all great curse ideas... but RAW-legal?)   Summation:  What are the limits of the 3rd lvl / Necromatic Bestow Curse spell for transformation? Rethrowing exceptions in Java without losing the stack trace As shown in various episodes ("Mirror, Mirror," "Phantasms," etc.), other universes exist, oftentimes containing the same characters, albeit in different situations than in the prime universe. I'm curious, though, whether individuals that apparently exist outside of the multiverse, such as members of the Q, would not be duplicated elsewhere. It seems reasonable to me, but I'm not sure. Why am I getting no power when I pedal? pm-hibernate freezes and/or fails to resume on Dell XPS 13, 14.04, 3.19 kernel I'm running 14.04 on a new XPS 13. In order to support the newer hardware, I'm running a mainline 3.19 kernel.   Hibernate resume has been iffy with the entire 3.19 kernel line, but recently (3.19.5) pm-hibernate no longer works at all. The screen will either a) switch off momentarily, then back on with all inputs frozen or b) switch off entirely, and fail to resume upon power on (despite displaying the correct resume-from-uuid message).  I have enough swap space (and it has worked in the past), I have checked that the uuid is being correctly referenced, and nothing seems particularly amiss in /var/log/pm-suspend.log (last 200 lines here: )  Thanks for your help. Transparent text cut out of background Let p be a prime. Consider the equation $\frac1x+\frac1y=\frac1p$. What are the solutions? Write down the set of distinct solutions and prove your list is complete. $x$ and $y$ are positive integers.  I have rewritten it as $\frac{(x+y)}{xy} = \frac1p$, but I don't understand where to go from here? Difference between sentences using 'was' and 'being'? What is the meaning of below sentences using was and being?     He committed suicide after he was severely beaten up by four teachers. He committed suicide after being severely beaten up by four teachers. map to drive on server with bat file I have a web server that I want to map a drive to it. I want to do this in a .bat file. Pseudo ->   Check if a drive at an ip address has been mapped to a local drive.  If it has then progress with a reference to the drive If not map to a local drive and then progress with a reference to the drive  Is this possible and if so how can I achieve it? I want to loop over the contents of a text file and do a search and replace on some lines and write the result back to the file. I could first load the whole file in memory and then write it back, but that probably is not the best way to do it.  What is the best way to do this, within the following code?  f = open(file) for line in f:     if line.contains('foo'):         newline = line.replace('foo', 'bar')         # how to write this newline back to the file When "etc." is at the end of a phrase, do you place a period after it? Example:     It's all about apples, oranges, bananas, etc.   VS.     It's all about apples, oranges, bananas, etc..   Update  What happens if the abbreviation is inside parentheses, do you place a dot after and before the closing parenthesis?     It's all about fruit (apples, bananas, etc.). Moving all files to a new computer, my concern is links I'm moving all files to a new computer, via an external hard drive. I am certain that broken links will be a concern. What can I do to fix this? Is there a 'Best Practices' kind of way to prevent this problem beforehand? I have many hundreds of .blend files, and of course, I fear losing them. Current sensing nA to mA I need to sense the current of a target  through various periods of its on/sleep/off state and then need to send this data back to another interface MCU to read out the value. They will be both on the same board, powered by 5&nbsp;V USB.  However these are the requirements I have:   5&nbsp;V USB PSU Must be able to sense currents in the range of nA to mA with high    accuracy/precision. (I figure 1&nbsp;nA to 500&nbsp;mA) Only measure the current of the target MCU and not the interface. Must output maximum 3.3&nbsp;V interface MCU   I have looked at available options and current sense amplifiers by Texas Instruments can't sense nA due to higher bias currents. So therefore I feel I would need a precision amplifier.  However I am stuck on how to proceed with this because, I need to somehow have a automatic dynamic current range and I am fairly new to current sensing and am not really sure on all the specifics. Refer to renamed columns in WHERE clause SELECT name, gdp/population as capita  FROM world w1  WHERE continent = 'Europe'       AND capita &gt;  (     SELECT gdp/population      FROM world      WHERE name = 'United Kingdom' );   How come I can't refer to capita in the WHERE clause? It returns this error:  Unknown column 'capita' in 'where clause'   I can however, run the query if I replace capita with gdp/population.  I don't want to do gdp/population &gt; .... How can I do capita &gt; ...? Or is this not possible? In the Monster Manual, gargoyles, golems, helmed horrors, and xorns all enjoy either immunity or resistance to:     bludgeoning, piercing, and slashing damage from nonmagical weapons that aren't adamantine   Is it actually possible to obtain a nonmagical adamantine weapon?   Although the Dungeon Master's Guide describes adamantine armor (p.150) and Daern's instant fortress (p.160) as being made of this metal, and provides an AC for destructible adamantine objects (p.246), there's no mention of adamantine weapons. The creation spell (PH p.229) can produce an adamantine weapon that lasts for a single minute. The true polymorph spell (PH p.283) can't turn a regular weapon into adamantine, but it might be able to  for an hour. The only way to obtain a (somewhat) permanent adamantine weapon in the game seems to be to loot one from drow elves (MM p.127). A PC (perhaps even a drow) with proficiency in smith's tools could presumably craft a nonmagical adamantine sword during downtime at 5gp per day (PH p.187), except that unlike silvered weapons, there's no pricing for adamantine.   It's seeming like mugging an NPC drow is the only by-the-book way to tool up against gargoyles. Is there an optimal loss function for dealing with imbalanced classes? I'm aware that there are many ways of dealing with datasets where there is a strong class imbalance in the target variable: downsampling the more prevalent and less important class, over-weighting the valuable minority class, etc. Are there any loss functions that are ideally suited for these types of problems? I've heard of AUC, but AUC sounds like a means of evaluating a model once its training is complete (correct me if I'm wrong). Log loss and squared loss penalize a model's over confidence, but overconfidence and class imbalance seem unrelated. Is that true? Prove $2^b-1$ does not divide $2^a + 1$ for $a,b>2$ I'm trying to prove $2^b-1$ does not divide $2^a + 1$ for $a,b&gt;2$. Can someone give a hint in the right direction for this? Ubuntu 14.04 slow wireless connection $A=\begin{pmatrix}0&amp;0&amp;1\\ 0&amp;1&amp;0\\ 1&amp;0&amp;0\end{pmatrix}$  How do I show that there are no solutions for $X^2=A$ in $M_3(R)$?  I tried $X^2=\begin{pmatrix}a_{11}&amp;a_{12}&amp;a_{13}\\ a_{21}&amp;a_{22}&amp;a_{23}\\ a_{31}&amp;a_{32}&amp;a_{33}\end{pmatrix}^2=A$, but I couldn't seem to find anything that shows there are no solutions. How should I approach this?