How to prove that if $\gcd(n,k)=1$ then $\gcd(f,nk)=\gcd(f,n)·\gcd(f,k)$ by primes
If $\gcd(a, b) = 1$, then $\gcd(ab, c) = \gcd(a, c) \cdot\gcd(b, c)$
Let $\alpha$ be a regular curve in $\mathbb{R}^2$ and let all of its tangent lines pass through the origin. Also, let $\beta$ be a regular curve in $\mathbb{R}^2$ and let all of its normal lines pass through the origin.   How can I show that $\alpha$ is contained in a straight line through the origin and that $\beta$ is contained in a circle around the origin?  I know that the tangent line $\alpha(t)$ is the line that points in the direction of the tangent vector $T(t)$ and that the normal line of $\beta(t)$ is the line that points in the direction of $N(t)$, but how can I put this all together to complete the proof? Prove that $ a $ divides $1^{a^{n}} + 2^{a^{n}} + ... + (a-1)^{a^{n}}$? Two finite abelian groups with the same number of elements of any order are isomorphic Suppose $G, H$ are finite abelian groups with the same number of elements of any given order. Show they are isomorphic.   Since finite abelian groups are isomorphic if and only if their Sylow subgroups are, we may restrict our attention to the case where $G, H$, are $p$-groups, but I can't quite make it past there... I'm working on a web application that needs to prints silently -- that is without user involvement.  What's the best way to accomplish this?  It doesn't like it can be done with strictly with Javascript, nor Flash and/or AIR.  The closest I've seen involves a Java applet.  I can understand why it would a Bad Idea for just any website to be able to do this.  This specific instance is for an internal application, and it's perfectly acceptable if the user needs to add the URL to a trusted site list, install an addon, etc. As far as I can tell, it's something like this:  damage reduction = armor / (armor + 100)   Is this accurate? Non-trivial cyclic group with unique generator Let $G$ be a non-trivial cyclic group with an unique generator. Show that $G$ has exactly two elements.   Let $g$ be a generator of $G$. My first idea to approach the problem was to show that if $|G|=n &gt; 2$ then there is an integer $k$ between 1 and $n$ such that $\langle g^k \rangle = G$ or, equivalently, $g \in \langle g^k \rangle$. My second idea was to analyse the problem in 2 separate cases. For the first one, if $n$ is prime, let $k$ be a positive integer then, by Lagrange's Theorem, $|g^k| \mid n$ and, therefore, $\langle g^k \rangle = n$. Hence, we must conclude that, if $n &gt; 2$ the generator cannot be unique. For the second case, where $n$ is composite, I was hoping to get to a contradiction, but could not advance any further. Is there a way to develop either of these ideas or is it necessary a new approach? C++ virtual function from constructor Why the following example prints "0" and what must change for it to print "1" as I expected ?  #include &lt;iostream&gt; struct base {    virtual const int value() const {       return 0;    }    base() {       std::cout &lt;&lt; value() &lt;&lt; std::endl;    }    virtual ~base() {} };  struct derived : public base {    virtual const int value() const {       return 1;    } };  int main(void) {    derived example; } How to select an element inside "this" in jQuery? How did Voldemort get his body back? In the first book it seems that Voldemort has been reduced to nothing but a ghostly wisp. But IIRC by the fourth book he somehow has a body that is carried around by Wormtail. In the first book he needs someone to inhabit, by the fourth, he can't really move around, but he does have a body back which can hold a wand.  How did this transition happen? Why does find -mtime +1 only return files older than 2 days? I am a beginner in learning selenium Webdriver. I am not able to understand how page object and factory models works and how it can be beneficial?    Kindly help me explain it with detailed explanation. What was the galaxy formation Rust saw before confronting Erol Childress in  S01E08?  It was weird like out of the series. I guess it was some sort of hallucination, a product of Rust's madness or addiction. I heard myself saying something was "tickety-boo", meaning good, successful, or satisfactory. Does anyone know where this strange-sounding phrase originated? What is the meaning of equation of a circle or equation of a cube? We have an equation of a Circle $$x^2 + y^2 = 1$$ and equation of a Cube $${||x-y|+|x+y|-2z|+||x-y|+|x+y|+2z|=1}$$  When I substitute x,y in first equation or x,y,z in the second equation then the equations are said to be satisfied when LHS and RHS are equal.  I have a quick question. If the first equation or second equation get satisfied, because LHS becomes equal to RHS, then those values of x and y of first equation are said to be lying on the circumference of the circle and not inside or outside of the circle and x,y,z of the second equation are said to be lying on the surface of the cube and not on the inside and not on the circle of the cube. Correct? Coloring arrows in \xymatrix Comparing two integers without any comparison Is it possible to find the greatest of two integers without any comparison? I found some  solutions:   if(!(a/b)) // if a is less than b then division result will be zero. {     cout &lt;&lt; " b is greater than a"; } else if (!(a-b)) // we know a is greater than or equal to b now.  check whether they are equal. {     cout &lt;&lt; "a and b are equal"; } else     cout &lt;&lt; "a is greater than b";   But if(c) or if(!c) is a comparison to zero. In addition it doesn't work for negative numbers. In fact I need a solution that avoids any if statement. Instead I should use switch statements and arithmetic operators. ThanX. What solutions does the equation 4x≡2mod10 have? (Hint, it will have more that one.) What about solutions to an equation ax≡dmodn, where d=gcd(a,n)? What solutions does the equation $4x\equiv2\mod10$ have? (Hint, it will have more that one.) What about solutions to an equation $ax\equiv d\mod n$, where $d=\gcd(a,n)$?  This is the question that appeared in our universities last year paper. But it seems very difficult for me like for first part I am able to solve by simplifications I got $x=5\cdot Z +3$ as solution which makes sense and seems fair but for the second one, I have no idea how can I approach that. Can you please help me out with that one. I tried to simplify that by using Euclidean algorithm but still, I didn't get any idea that what actually the solution will be. My final exam is on 2nd of next month and because of covid I am unable to contact in person for any help. So any help from this site will be appreciated. Thank you for looking at this question! In a Cartesian system, I've got the slope, start point and distance of a line segment.  What's the formula to find the endpoint? Bmp to jpg/png in C# My issue is that I'm trying to write variable names in italic, but as some of them have underscores in their names, LaTeX assumes that I am writing an equation and it messes up the rest of my document after that point. Here is a sample of my code: As variable \textit{sensor.linear_acceleration} is a matrix  The underscore in linear_acceleration is what makes my whole document math modded, and is a matrix renders as if it was an equation. How can I display that underscore in italic without LaTeX making everything an equation?