This is a somewhat obvious fact that is intuitively obvious to me, but I haven't been able to construct a proof of it. Euclid's lemma states for for $p$ a prime and $ab$ a product of integers (let's take everything to be positive for simplicity), if $p \mid ab$, then $p \mid a$ or $p \mid b$. This is clear, and I know how to prove it. Let's extend it somewhat. Suppose that $a$ and $b$ are two relatively prime integers, and we have $a \mid bc$ for some other integer $c$. Then $a \not \mid b$, so it must divide $c$. This fact is obvious to me, but I can't figure out how to prove it. Does anyone have any hints or advice? Do I need the assumption of positivity? (For my purposes at the moment, I only need them to be positive, but there is value in having the most general result possible). EDIT: Updated attempt: We have that $a,b$ are relatively prime, so there exist $r,s \in \mathbb{Z}$ such that $ar + bs = 1$ by Bézout's lemma. Multiply through by $c$ to get $arc + bsc = c$. Then $a \mid arc$ and $a \mid bsc$, so $a \mid c$. How is that?
Let $A$ be a UFD. Assume that $a,b \in A$ are relatively prime, $c \in A$ and $a | bc$. To prove that $a|c$, is the following approach correct (or do you have to use some type of prime factorization argument)?  By the relatively prime assumption, $\gcd(a,b)$ is a unit. Call that unit $u \in A^{\times}$. Then we can write $u = ax + by$ for some $x,y \in A$. But then, $c = c \cdot u = c(ax+by)=cax+cby$. However, it is clear that $a|cax$ and the fact that $a|bc$ implies $a|cby$. Thus, $a|cax+cby=c$.
Suppose that I have a &lt;div&gt; that I wish to center in the browser's display (viewport). To do so, I need to calculate the width and height of the &lt;div&gt; element.   What should I use? Please include information on browser compatibility. Proof verification: Set of partial limits of a sequence is closed Difference in output between SAS's proc genmod and R's glm I'm trying to replicate a model fitted in SAS in R but the fit I'm getting gives me slightly different coefficients and standard errors.  Data:  testdata &lt;- data.frame(matrix(c("f","Test", 1.75,   16, 0,  16, 0,  1,  1, "m",    "Test", 1.75,   15, 1,  16, 6.25,   1,  0, "f",    "Test", 2.75,   4,  12, 16, 75, 1,  1, "m",    "Test", 2.75,   9,  6,  15, 40, 1,  0, "f",    "WHO",  1.75,   15, 1,  16, 6.25,   0,  1, "m",    "WHO",  1.75,   14, 2,  16, 12.5,   0,  0, "f",    "WHO",  2.75,   2,  13, 15, 86.6667,    0,  1, "m",    "WHO",  2.75,   3,  13, 16, 81.25,  0,  0 ), ncol=9, byrow=TRUE)) names(testdata) &lt;- c("sex", "vaccine", "dose", "not_p", "para", "n", "pct",                       "vacnum", "sexno")   SAS:  proc genmod data=model_data; class sex; model para/n  = dose sex vacnum    /dist=bin    link=logit   type3; run;  Analysis Of Maximum Likelihood Parameter Estimates  Parameter   DF Estimate Std Error Wald 95% Conf Lim Wald Chi-Square Pr &gt; ChiSq  Intercept   1 -9.4020   1.6220    -12.5810  -6.2230  33.60           &lt;.0001  dose        1  3.9208   0.6460      2.6546   5.1870  36.83           &lt;.0001  sex f       1  0.5574   0.5184     -0.4587   1.5735   1.16           0.2823  sex m       0  0.0000   0.0000      0.0000   0.0000    .              .  vacnum      1 -1.3221   0.5483     -2.3967  -0.2475   5.81           0.0159  Scale       0  1.0000   0.0000      1.0000   1.0000        R:  testdata$sexno &lt;- as.factor(testdata$sexno)     a &lt;- contr.treatment(2, base = 1, contrasts = TRUE)  contrasts(testdata$sexno) &lt;- a  fitreduced &lt;- glm(para/n ~ dose + as.factor(sex) + vacnum,                    family=quasibinomial(link="logit"), data=testdata)  coef(summary(fitreduced))                    Estimate Std. Error   t value    Pr(&gt;|t|) (Intercept)     -9.4013750  1.7613982 -5.337450 0.005935450 dose             3.9173794  0.7001133  5.595351 0.005007179 as.factor(sex)1  0.5704671  0.5568436  1.024466 0.363525300 vacnum          -1.3336100  0.5887552 -2.265135 0.086189704   I believe I have the right contrasts to give me a type III SS but there is a small discrepency in values, have a missed something here? Figuring out algorithm/formula for item scaling in WoW Opportunity attack when moving in and out of range of a creature in the same move? Let's say I'm 10ft away from an enemy. In my turn, I choose to move 5ft to get close enough to attack, I do attack, then I use my remaining movement to move out of range of the enemy. Does this still allow them an opportunity attack? Every partially defined isometry can be extended to a isometry I know that the following theorem holds true:  Let $S$ be a subset of $\mathbb R^n$, and let $f:S\to \mathbb R^n$ a map such that $d(p,q)=d(f(p),f(q))$ for every $p,q \in S$ (here $d$ is the usual euclidean distance). Suppose that $0 \in S$ and $f(0)=0$. Then $f$ can be extended to an orthogonal transformation of $\mathbb R^n$.  I have two questions:   Can someone give me a reference of a proof of this theorem? I believe that the previous theorem holds true also if we change $\mathbb R^n$ with $\mathbb C^n$ and "orthogonal" with "unitary", with nearly the same proof. Does it holds true also in the $ \ell^2$ space? With the same proof? What is the order of $(\mathbb{Z} \oplus \mathbb{Z})/ \langle (2,2) \rangle$ and is it cyclic? Evidently, $(\mathbb{Z} \oplus \mathbb{Z})/ \langle (2,2) \rangle$ has order $4$, but I think it's infinite.   The four cosets are listed as $(0,0) +  \langle (2,2) \rangle$, $(0,1) +  \langle (2,2) \rangle$, $(1,0)+  \langle (2,2) \rangle$ and $(1,1) +  \langle (2,2) \rangle$. However, $(2,0)$ doesn't appear to be in any of these cosets. Maybe the answer I'm being told is wrong. Why do close votes expire? I nominated  to be closed as a dup. I see that two other people upvoted my comment, so they might have voted to close it as well. Now, we didn't gather enough votes, and all close votes expired ... and I can't re-vote to close. What is the motivation behind expiring close votes ... especially for close-as-duplicate? Update: Please take into consideration:  Beta Stack Exchange sites have a smaller community, and fewer people with voting privileges Closing as a duplicate is &quot;usually the right thing&quot; IMHO - much less controversial than closing as off topic. After a vote is reset, the original users can't vote again. On a small site, this makes it even harder to get a consensus of five votes. Is it better in C++ to pass by value or pass by constant reference? Count number of increasing functions, nondecreasing functions $f: \{1, 2, 3, \ldots, n\} \to \{1, 2, 3, \ldots, m\}$, with $m \geq n$. I stumbled upon a question given like:  Let $m$ and $n$ be two integers such that $m \geq n \geq 1$.   Count the number of functions $$f: \{1, 2, · · · , n\} \to \{1, 2, · · · , m\}$$ of the following two types:  (a) strictly increasing; i.e., whenever $x &lt; y, f(x) &lt; f(y)$, and  (b) non-decreasing; i.e., whenever $x &lt; y, f(x) \leq f(y)$.  I tried in the following way.  For (a). We can say $f(n)&gt;f(n-1)$ AND $f(n)&gt;f(n-2) \ldots f(n)&gt;f(1)$ (total $n-1$ elements)  Similarly for $f(n-1)&gt;f(n-2), f(n-1)&gt;f(n-3) \ldots f(n-1)&gt;f(1)$ (total $n-2$ elements)...  and $f(2)&gt;1$  Like this $$(n-1) + (n-2) + \ldots + (1) = \frac{n(n-1)}{2}$$  Is this correct?  For (b). Since there's equality, it will be $$n + (n-1) + ... (1) = \frac{n(n+1)}{2}$$  Is my approach correct? Any help is appreciated. Thanks in advance. How do I get rid of the noise on my Hayes Stroker Trail brakes Sometimes, when fighting an enemy, your character will do a special move, with a third person view animation, instantly killing the enemy. But these types of moves seem to be random, sometimes being used on enemies with nearly full health, other times not having a finishing move at all during a whole fight.  What are the parameters determining when a finishing move is used? how to use common modulus attack? I am struck with the following problem:  Let Alice, Bob, Chris and Eve communicate over a public network. They encrypt all messages they send using RSA system. Bob and Chris have the RSA modulus $n_B$ and $n_C$ respectively with $n_B$ = $n_C$ But different public encryption exponents: $e_B \neq e_C$. Suppose $\gcd(e_B, e_C)=1$, and that Alice sends the same secret message to Bob and Chris. Show how Eve can decipher the message. Are confidence intervals open or closed intervals? After writing code to populate textboxes from an object, such as:  txtFirstName.Text = customer.FirstName; txtLastName.Text = customer.LastName; txtAddress.Text = customer.Address; txtCity.Text = customer.City;   is there way in Visual Studio (or even something like Resharper) to copy and paste this code into a save function and reverse the code around the equal sign, so that it will look like:  customer.FirstName = txtFirstName.Text; customer.LastName = txtLastName.Text; customer.Address = txtAddress.Text; customer.City = txtCity.Text; Customize Ubuntu for a library Internet kiosk Saving Arduino sensor data to a text file How can I save data retrieved from a sensor to a text file on the computer? What is causing disabled options when adding a mesh/curve object to the scene? Sometimes when adding an object to the scene I see that all options (on toolbar or in window after pressing F6) are disabled. Can anyone tell me when can this happen? What is causing it and how to revert it - make it enabled again. It worked for me when I restarted Blender and reloaded the scene. However I would really like to know what can cause it? How do I enable Remote Desktop connection on Xubuntu 11.10? Is it possible to set up and watch a game between just computer AI opponents? No human players, just AI and have the ability to scroll around and see what the AIs are doing with the turn management being also automated, so the AIs just end turn when they are done. Oh, and all of this hopefully without any sort of fog of war.  This way, I can see how the AI manages resources and units and maybe learn a thing or two as a civ newb.