The discussion at  suggests that all common multiples of a and b are multiples of $lcm(a,b)$ and this seems to be true. Certainly all multiples of $lcm(a,b)$ are multiples of a and b, but how do I show that there are no others?
Is $$d_1\mid n,d_2\mid n\iff [d_1,d_2]\mid n$$ true ? And if yes, how can I prove it ? I recall that $[d_1,d_2]$ is the least common multiple.  My tries  For the implication : Let $d_1\mid n$ and $d_2\mid n$. Then, $d_1d_2\mid n^2$. By the way, since $d_1,d_2\mid n$, we have that $(d_1,d_2)\mid n$ where $(d_1,d_2)=:\gcd(d_1,d_2)$  and thus $$[d_1,d_2](d_1,d_2)\mid n^2.$$  Question: How can I get $[d_1,d_2]\mid n$ from this?  For the converse, since $d_1,d_2\mid [d_1,d_2]\mid n$, the claim follow.
Does anyone know where I can find cheat codes for Goldeneye 007 Wii? I know that the game hasn't even been out for a week, but does anyone know if any cheat codes have been released by Activision yet?  I know that they must exist since there is a cheat codes menu within the game. Class not found loading JDBC org.postgresql.Driver Using a naked black hole as a mirror? How to generate random points on a sphere? Order of a homomorphism of groups Let $f: G \to H$ be a homomorphism of groups. Assume that $a\in G$ and $\operatorname{ord}(a)=n$. Prove that the order of $f(a)$ is a divisor of $n$. I know that if $H$ is a subgroup of $G$, then $$|G|=|H|\cdot|\text{distinct cosets of $H$}|$$ so $|H|$ must be divisor of $|G|$, but I don't understand how to apply that logic to homomorphisms (or if that will even work). Any help is appreciated! Misplaced Accents in Cyrillic Text In educational material for the Russian language and linguistic literature the need to put accents on Cyrillic vowels often arises. However when I tried to do this with luaLaTeX the accents where misplaced.  \documentclass[11pt]{article} \usepackage[utf8]{inputenc} \usepackage{fontspec} \setmainfont{Linux Libertine:style=Regular} \begin{document}  Е́сли вы изу́чите  \'{Е}сли вы из\'{у}чите  \'{E}сли вы из\'{y}чите % Here "E" and "y" are Latin glyphs, but this workaround does not                         % work with all Cyrillic letters e.g. Я, Ю \end{document}     The last line would looks nice, but unfortunately these are Latin glyphs.  Is this a problem with the font used (Linux Libertine) or with LaTeX? Best way to recover mysql from a "chmod -R 777 /" with databases intact Splitting lines by polygons in QGIS I have a river network (lines) and a .shp with states (polygon). I would like to split the lines at the state boundaries i.e. split lines by polygon boundaries. Is there a geoprocessing tool to do this in QGIS? I can't merge the lines by the way, since they are meaningfully split into tributaries/main river. Do votes get canceled if voting privilege is lost? If I'm below 15 reputation, I can't alter the public score of an answer/question.  Suppose I gain the privilege by getting the required rep and vote some posts up. Then, for any reason, I fall below 15 and lose the ability to vote. What about the votes I already did cast?  Do post scores get altered by ignoring my vote, or will they stay the same? In other words, is losing a privilege retroactive?  If yes, what about when I gain my privilege back? Are my votes automatically restored, or do I have to cast them again from scratch?  I made the easiest example, but this applies to any higher privilege link new object to scene with python in 2.8 In Python 2.8, I would like to create copies of the object 'CubeTemplate' but am having no luck.  I have reviewed every blender.stackexchange post on this subject (, ,  and ) but cannot get them to work - either they don't apply or the sample code throws errors (perhaps because it was pre 2.8).  The code below throws this error:       AttributeError: 'bpy_prop_collection' object has no attribute 'link'   for x in range(1, 3):     bpy.data.objects['CubeTemplate'].select_set(True)     bpy.context.view_layer.objects.active = bpy.data.objects['CubeTemplate']     me = bpy.context.object.data # use current object's data     me_copy = me.copy()     ob = bpy.data.objects.new('CubeCopy'+str(x), me_copy)     ob.location = (0, 0, x)     scene = bpy.context.scene     scene.objects.link(ob)     scene.update() How do I pre-populate a jQuery Datepicker textbox with today's date? I have a very simple jQuery Datepicker calendar:  $(document).ready(function(){     $("#date_pretty").datepicker({      }); });   and of course in the HTML...  &lt;input type="text" size="10" value="" id="date_pretty"/&gt;   Today's date is nicely highlighted for the user when they bring up the calendar, but how do I get jQuery to pre-populate the textbox itself with today's date on page load, without the user doing anything?  99% of the time, the today's date default will be what they want. [S]he has the ears of a ...? Often, when overheard from far away, I find myself saying/thinking:     [S]he has the ears of a hawk!   Which doesn't really make sense as hawks aren't particularly well known for their sense of hearing.  Is there a common saying, equivalent to "eyes of a hawk", relating to someone's sense of hearing? Linear regression with non-symmetric cost function? I want to predict some value $Y(x)$ and I am trying to get some prediction $\hat Y(x)$ that optimizes between being as low as possible, but still being larger than $Y(x)$. In other words: $$\text{cost}\left\{ Y(x) \gtrsim \hat Y(x) \right\}  &gt;&gt; \text{cost}\left\{ \hat Y(x) \gtrsim Y(x) \right\}  $$  I think a simple linear regression should do totally fine. So I somewhat know how to implement this manually, but I guess I'm not the first one with this kind of problem. Are there any packages/libraries (preferably python) out there doing what I want to do? What's the keyword I need to look for?  What if I knew a function $Y_0(x) &gt; 0$ where $Y(x) &gt; Y_0(x)$. What's the best way to implement these restrictions? How do I use sudo to redirect output to a location I don't have permission to write to? I've been given sudo access on one of our development RedHat linux boxes, and I seem to find myself quite often needing to redirect output to a location I don't normally have write access to.  The trouble is, this contrived example doesn't work:  sudo ls -hal /root/ &gt; /root/test.out   I just receive the response:  -bash: /root/test.out: Permission denied   How can I get this to work? Is "education" a count noun? Reading shapefile using C#? Problem with finding the GCD in the Field of Polynomials mod p I am really having trouble working a problem in my course. I will post it and then add what I have thought etc.  Let p be an odd prime, then using the Euclidian Algorithm, calculated the GCD of of $x^{p-1}-1$ and $x^2+1$ in $\mathbb{Z} / p\mathbb{Z}[x]$  Also, how does $p=1 (mod4)$ tell us that  $-1$ is a square in $\mathbb{Z} / p\mathbb{Z}$, infact that $-1$ is a square here if and only if $p=1(mod4)$.  My thoughts:   I know that $$x^{p}-x=(x)(x-1)(x-2)…(x-(p-1))$$ and so that in this sense our roots will be the congruence classes of $1,2,…p-1$  and also that $$x^{p-1}-1=(x-1)(x-2)…(x-(p-1))$$  But I get lost when trying to apply the EA to these polynomials,  like how can I do it when p is arbitrary?  Preferably id like to be able to understand/see how to do the first part first i.e. using division with residue and then move on to the second part. Any insight/hint or solution to this would be great,  Thank you all  Update:  When I try to do division with residue, I just get  $$x^{p-1}-1=(x^2+1)(x^{p-3}-x^{p-5}+x^{p-7}-x^{p-9}...)$$  for example,  $p=7$  $$(x^{6}-1)=(x^2+1)(x^4-x^2+1)$$  for $p=11$  $$(x^{10}-1)=(x^2+1)(x^{8}-x^{6}+x^{4}-x^{2}+1)$$  which for specific cases tells me that 1 is the gcd, but I dont know how to generalize that for the case with arbitrary prime p. Why was the snaphat not awarded? Or why was it awarded incorrectly? There have been a few cases in which the  was not awarded.   There have also been other cases in which it was awarded incorrectly, on old questions.   What gives? Let R be a commutative ring with an identity such that for all $r\in$ R, there exists some $n&gt;1$ such that $r^n = r$. Show that any prime ideal is  maximal. (Atiyah and MacDonald, Introduction to Commutative Algebra, Chapter 1, Exercise 7.)    Any hints? How do I completely reset network settings?