$8x\equiv 21\pmod{24}$  I am having problem solving this linear congruence because 8 is even and 21 is odd.
For example: $$7x \equiv 1 \pmod{31} $$ In this example, the modular inverse of $7$ with respect to $31$ is $9$. How can we find out that $9$? What are the steps that I need to do? Update If I have a general modulo equation: $$5x + 1 \equiv 2 \pmod{6}$$ What is the fastest way to solve it? My initial thought was: $$5x + 1 \equiv 2 \pmod{6}$$ $$\Leftrightarrow 5x + 1 - 1\equiv 2 - 1 \pmod{6}$$ $$\Leftrightarrow 5x \equiv 1 \pmod{6}$$ Then solve for the inverse of $5$ modulo 6. Is it a right approach? Thanks.
Can't convert string into float/double How to find stopping distance of a car? Pluralizing abbreviations where the noun is not the last word For abbreviations - both acronyms and initialisms - where the last word in the abbreviation is the noun, I'm accustomed to adding an 's' with no apostrophe (e.g. ATMs), as described in the answer to this question:   But how do you pluralize abbreviations where the noun is not the last word? E.g. 'Point of Presence' (POP)  I can see how 'POPs' could read 'Point of Presences', when the intention is 'Points of Presence.' I  want to download free app in my iPod touch device and i have the following doubts.  Kindly resolve me doubts.   when i try to download it asks my credit card security number. since i am trying to download free app , why do i need to provide my credit card security code? Will i be debited for downloading the free app from app store?   I cant able to find any information regarding this on apple.  Please provide me some references about this from apple and please resolve my doubts.  Thanks a lot. Scaling HUD with variable resolution Naming threads and thread-pools of ExecutorService Want to copy the file name quickly in Windows. I always use rename, copy , but feel it is quite slow. Any better way? Limit of $s_n = \int\limits_0^1 \frac{nx^{n-1}}{1+x} dx$ as $n \to \infty$ Let $s_n$ be a sequence defined as given below for $n \geq 1$. Then find out $\lim\limits_{n \to \infty} s_n$.   \begin{align}   s_n = \int\limits_0^1 \frac{nx^{n-1}}{1+x} dx  \end{align}  I have written a solution of my own, but I would like to know it is completely correct, and also I would like if people post more alternative solutions. Format a number with optional decimal places in Excel I have numbers in cells in Excel. I want the numbers formatted so that if they have decimal places they show to a maximum of two, and if they have no decimal places it doesn't show any.  For example.    15 should be formatted as 15, NOT 15.00 14.3453453 should be formatted as 14.35 12.1 should be formatted as 12.1  0 should be formatted as 0   The closest custom format code I've come up with is 0.##. Unfortunately this formats 15.00 as 15. (note the extra decimal point).  To further complicate the issue, the spreadsheet is a result of an export from SQL Server Reporting Services. So no macros are possible. Oh well, it looks like 0.## is my best bet, and they can just live with the extra period. Cruise ship in the Schengen area: Do the days spent at sea count toward the limit for short visits? I am planning on spending 90 days in Schengen zone countries.  I will be taking a cruise between Lisbon and Barcelona.  Do the days at sea count against the 90 days allowed for staying in the Schengen zone? What’s the point in having “www” in a URL? Other than for historical reasons, is there is reason to have “www” in a URL?  Should I create a permanent redirect from www.xyz.com to xyz.com, or from xyz.com to www.xyz.com? Which one would you suggest and why? Which is better: ... or ... I need to delete some rows from one large table. The rows to delete shouldn't be in another table, example:  DELETE FROM LargeTable WHERE id IS NOT IN (SELECT DISTINCT foreign_id from EvenLargerTable)  but my server can't handle such blunt query, because there are almost a million records in the LargeTable and few million records in the EvenLargerTable  How can I solve it? When I get a third party app it says it needs to be verified before I can use it but when I go to settings to verify I tap verify and nothing happens. Is there something I can do so I don’t have to constantly reinstall my apps? 'was' versus 'had been' When I asked the driver/ to come with me to the temple/ he had been reluctant./ No error.   I think instead of 'had been', there should be 'was'. Even the answer key points that the error is in the 'he had been reluctant' part. But I am not sure why exactly is 'had been' wrong.  If we replace the first part with 'when I had asked the driver', would that make 'had been' in the third part correct? Changing data type in ArcGIS attribute table? Increase the visibility of the association bonus by documenting it with the other privileges I followed @Micha's instruction and successfully used the modules in GRASS. I use r.watershed to get the basin raster and the drainage line：  and I use r.water.outlet to get the watershed of one outlet：  I got the whole basin area of the green outlet point, but all I want is the little part of the original basin area, like this：part B is what I want  This schematic photo shows the idea of what I want：  Can I achieve this goal using r.water.outlet or other modules in GRASS？ Or all I can do is using the intersect tool in Geoprocessing tools in QGIS？  I want to find the watershed area, but using QGIS. I have following files to work with：  stream network shapefile(polyline) point shapefile clipped from the polyline shapefile a whole island(the stream is inside the island) DEM(I got 2 types:.lan &amp; .tiff)  I want to find the total watershed area located upstream of every point in the network, here's the schematic diagram：  the green, red, and yellow line enclose the watershed of each point(here I only take 3 points as the example), and I want to calculate their area. I already got the DEM file, and I must cut the region that contains the stream off, should I polygonize it first？ Can I directly use this form(.lan or .tiff) to work with GRASS？ A inequality proposed at Zhautykov Olympiad 2008 An inequality proposed at Zhautykov Olympiad 2008.     Let be $a,b,c &gt;0$ with $abc=1$. Prove that:    $$\sum_{\mathrm{cyc}}{\frac{1}{(a+b)b}} \geq \frac{3}{2}.$$   Set $a=\frac{x}{y}$, $b=\frac{y}{z}$, $c=\frac{z}{x}$.   Our inequality becomes: $$\sum_{\mathrm{cyc}}{\frac{z^2}{zx+y^2}} \geq \frac{3}{2}.$$  Now we use that: $z^2+x^2 \geq 2zx.$ $$\sum_{\mathrm{cyc}}{\frac{z^2}{zx+y^2}} \geq \sum_{\mathrm{cyc}}{\frac{2z^2}{z^2+x^2+2y^2}} \geq \frac{3}{2}.$$  Now applying Cauchy-Schwarz we obtain the desired result.  What I wrote can be found on this link: . But now, I don't know how to apply Cauchy-Schwarz.   Thanks:) Install Ubuntu from ISO image directly from hard disk of a system running Linux?