I am looking for other solution of a problem, which I named: $$\text{Simple hit problem}$$ Let: $$y = \frac{a x + b}{c}$$ $$a, b, c, x \in \Bbb{Z}$$ For which $a, b, c$ there exists $x$ for such $y \in \Bbb{Z}$ ?  I noticed that the solution exists for $a, b, c$, if: $$(ax) \ \text{mod} \ c = c - (b \ \text{mod} \ c),$$ but the approach relies on checking all unique values of result of modulo operator from left side of equation.  The question is, if there exists any better approach for this problem or anyway it come down to check, if some equation will be true for some $x \in A$, where A will be a set of finite ammount of values ?
Isn't finding the inverse of $a$, that is, $a'$ in $aa'\equiv1\pmod{m}$ equivalent to solving the diophantine equation $aa'-mb=1$, where the unknowns are $a'$ and $b$? I have seem some answers on this site (where the extended Euclidean Algorithm is mentioned mainly) as well as looked up some books but there is no mention of this. Am I going wrong somewhere or is this a correct method of finding modular inverses? Also can't we find the Bézout's coefficients by solving the corresponding diophantine equation instead of using the extended Euclidean Algorithm?
Proof of "Dual normed vector space is complete" How would I go about finding $\prod_{n=1}^\infty (1+\frac{1}{n^2})$ ? Setting $a_k = \prod_{n=1}^k (1+\frac{1}{n^2})$, the first few partial products are $a_1 = 2$, $a_1 = 2.5$, $a_3 \approx 2.78$, $a_4 \approx 2.591$, $a_{20} \approx 3.50$, and they eventually appear to converge at around $3.639$. (These values were found with a quick python program. So much for my initial conjecture that the product converged to $e$.)   I tried writing $1+\frac{1}{n^2}$ as $\frac{n^2 + 1}{n^2}$, but it wasn't immediately obvious how to prove convergence / find the limit because the product doesn't really factor or telescope. Is there a way to calculate the limit? How to get product recommendations Possible Duplicate:          I have a question of the form "what companies provide service X?"  I checked meta, and apparently such questions are highly discouraged on all Stack Exchange sites. Which is fine, but it obviously raises the meta-question: Where can I ask my question?  (In particular, I know that there are people hanging out on Stack Overflow who have probably needed the same service, so if an answer exists, they possess it...)  Edit: I should perhaps point out that the service I'm looking for is rather specific and slightly obscure. Otherwise, yeah, this would be a 10-second exercise with Google. Just freeze an application (suspend the CPU, save its state to disk) and 'defrost' it when I want to use it 4 weeks later. I want to make some 3d models of pixel art. I started by adding a plane, adding a texture to it, added loop cuts to my plane, 15 loop cuts because it's a 16x16 texture, then I selected the blank faces and deleted them. As I extrude this plane, A &gt; E &gt; $0.1$, the texture messes up on the edge of the item, as shown below  Although not all faces mess up on the edge, a lot still do. I tried recalculating normals, unwrapping the object again, but I ended with the same problem, no change at all. Though I may have done it wrong, since I never did it before. I tried smart UV project and cube projection, but they just mess it up even more. Does anyone have a solution to this? Browsing folders with more than 100000 images PDE separation of variables Hi could someone guide me this problem  It says ,  $ u_t - u_{xx}-2 u_x=0  $  Use the method of separation of variables to find all possible solutions.   Could someone help me out for this problem. I'm beginner at PDE. I would be much appreciated if you able to show some partial work so that I can understand.  Thanks in advance for taking my consideration Is there a correct gender-neutral singular pronoun ("his" vs. "her" vs. "their")? Is there a pronoun I can use as a gender-neutral pronoun when referring back to a singular noun phrase?  Each student should save his questions until the end. Each student should save her questions until the end. Are used Schengen/UK visas acceptable for Turkish e-visa How can I list installed packages by installation date?  I need to do this on debian/ubuntu. Answers for other distributions would be nice as well.  I installed a lot of stuff to compile a certain piece of code, and I want to get a list of the packages that I had to install. Test the series whose general term is: $$\sqrt{n^2 + 1} - n$$   This is from Higher Algebra by Henry Sinclair. I'm struggling to understand how this series is divergent. If you do the following expansion: $$\sqrt{n^2 + 1} - n$$ $$= n\biggl(1 + \frac{1}{n^2}\biggr)^\frac{1}{2} - n$$ $$= n + \frac{1}{2n} - \frac{1}{8n^3} + ... - n$$ $$= \frac{1}{2n} - \frac{1}{8n^3} + ... $$  The remaining series appears to converge to zero as $n$ approaches infinity.  Their explanation is similar to mine, but at the end they say that the above series approaches $\frac{1}{2n}$. From there they directly say that the series is divergent. Earlier in the chapter, it's proved that the expansion of $(1 + x)^n$ is convergent for $x &lt; 1$. In this case, $\frac{1}{n^2} &lt; 1$ as $n$ approaches infinity, so I would assume that the series converges.  The solution to the problem is given  (number 17)  Any help is appreciated. Thanks. Can I use a Regex in an XPath expression? Something like .//div[@id='foo\d+] to capture div tags with id='foo123'.  I'm using .NET, if that matters. Show that if the smallest prime factor $p$ of the positive integer $n$ exceeds $\sqrt[3]{n}$, then $\frac{n}{p}$ must be prime or 1.  I'm stuck trying to prove this. I tried this using contradiction, but I haven't been able to prove it.  My approach:  Let $p$ be the smallest prime factor of the positive integer $n$.  Let $p &gt; \sqrt[3]{n}$  Then we must show that $\frac{n}{p}$ is either a prime number or 1.  We know that $n = pe$ for some $e \in \mathbb{Z}$.  Thus, $\frac{n}{p} = e$  We also know that $p$ > $\sqrt[3]{n}$, so $p^3 &gt; n$, and $p^2 &gt; \frac{n}{p}$  So, assume $\frac{n}{p} = e$ is not prime nor 1, meaning it's a composite number. Let's try to get a contradiction.  If $\frac{n}{p}$ is composite, then $\exists a,b \neq 1: \frac{n}{p} = ab $  Now, here's where I get stuck. What can I do with this info? $\frac{n}{p} = ab $ and $p^2 &gt; \frac{n}{p}$. How to disable the skype ads (advertisement)? Show that $\frac{(2n)!!}{(2n+1)!!}$ converges. How would I go about proving this limit?  $\lim_{x\to\infty}\frac{1}{x^4+x^2+5}=0$  so far I have:  $|f(x) -L|&lt; ϵ$ wherever $x &gt; N$  $|\frac{1}{x^4+x^2+5} - 0| &lt; ϵ $ wherever $x &gt; N$  $|\frac{1}{x^4+x^2+5}| &lt; ϵ  ,\; x &gt; ∞$, assuming, $x &gt; 0$  taking the absolute value $\frac{1}{x^4+x^2+5} &lt; ϵ$  ${x^4+x^2+5}$ > $\frac{1}{ϵ}$  i am not sure whether this is correct, but what would be the next step? Is "aren't I" correct grammar? Generalizing the case $p=2$ we would like to know if the statement below is true.  Let $p$ the smallest prime dividing the order of $G$. If $H$ is a subgroup of $G$ with index $p$ then $H$ is normal. Will installing Ubuntu using the "erase entire disk" option remove and erase all currently existing partitions? I'm considering ditching Windows 8 and doing a fully clean install of Ubuntu. However, I'm concerned because my computer came with a built-in 16 GB recovery partition that would need to be erased as well.   If I pick the "erase entire disk" option during installation, will it wipe out this recovery partition too? Why didn't the Hobbits get frostbite?