Prove or disprove 'If $\gcd(a,b)=1$ then, $\gcd(a^2,b^2)=1$, with $a,b\not= 0$'  I need to prove this statement. I think it is true and also the converse is true.  I took some examples such as $\gcd(2,5)=1$, and $\gcd(4,25)=1$...  But how can I prove this? I know that $1=ax+by$ for some $4x,y\in \mathbb{Z}$.. But not sure   How to go from there..   Any clue?
Show that if $a$ and $b$ are positive integers with $(a,b)=1$ then $(a^n, b^n) = 1$ for all positive integers n  Hi everyone, for the proof to the above question,  Can I assume that since $(a, b) = 1$, then in the prime-power factorization of a and b, they have no prime factor in common, when they are taken to the $nth$ power, they will still have no prime factors in common, and so $(a^n, b^n) = 1$ for all positive integers n.  I think I'm jumping to conclusions here again, if so, leave some tips on how to do the proof properly, thanks :)  And also, I do not know how to approach the reverse problem where if $(a^n, b^n) = 1 then (a, b) = 1$, any guidance will be much appreciated!
I'm trying to determine the best way to truncate or drop extra decimal places in SQL without rounding.  For example:  declare @value decimal(18,2)  set @value = 123.456   This will automatically round @value to be 123.46, which is good in most cases.  However, for this project, I don't need that.  Is there a simple way to truncate the decimals I don't need?  I know I can use the left() function and convert back to a decimal.  Are there any other ways? What is the difference between CurrentCulture and CurrentUICulture properties of CultureInfo in .NET? In .NET there is the CultureInfo class in the System.Globalization namespace. It has two similar properties both returning values of the CultureInfo type: CurrentCulture and CurrentUICulture.  What is the difference between them?   Which one should I use when and why? Obviously you need to have some high level idea of what's going on in the world around the players, and where you are heading with the story, but in an ongoing game at some point players will be players and will head off in a different direction to what you planned.  How far ahead should you be planning to minimise or eliminate being caught off guard, but also try to minimise the amount of 'wasted content' you come up with but never gets used? past simple or present perfect Please have a look at this sentence:     We haven't gotten a lot of Finnish groups that we liked here at Bananas but I'm pretty happy with this four tracker from Helsinki, apparently recorded in the same place where the Ramones played in the 70's.     The first verb is in present perfect (up to now) and the second is in simple past tense. I don't understand the simple past for the second verb.  Why "like" is not in its present perfect form as the first verb is? Should black holes be considered two-dimensional or three-dimensional? How do I access the list of currently logged on users through Terminal Services Manager in Windows Server 2012? Let $G$ be a group and $\{H_i\}_{i\in I}$ a family of subgroups. I would like to find a condition that will imply that $\cup_{i\in I} H_i$ is a subgroup. I know that it's not true in general, I need help to find this condition.  Any help is welcome.  Thanks a lot. Find the maximum value of $ f(x) = \int^{x}_{0} \sqrt{(x^2-x)^2+y^4}~dy$ Is there a tool to get the lines in one file that are not in another? Is there any tool that can get lines which file A contains, but file B doesn't? I could make a little simple script with, e.g, perl, but if something like that already exists, I'll save my time from now on. Connection pooling in PHP Is it possible to cache database connections when using PHP like you would in a J2EE container?  If so, how? C array setting of array element value beyond size of array I just bought a new house and I'm looking to run some cables. One of the areas I would like to run it to would require drilling through a 2x10 sistered joist. Realizing that going through the joist should be a last resort, if the hole is 1/4" and in the middle of one of the joists will this weaken the integrity?  I have seen lots of articles about drilling through the depth of a joist, but very few regarding drilling vertically. Remove sudo privileges from a user (without deleting the user) Do you pluralize the singular possessions of / items or people associated with individual members of a plural group? In a situation where say a group (or at least a plurality) of men is being addressed — for example on a sign passed by many married men — which is correct?  &quot;Remind your wife.&quot;  or  &quot;Remind your wives.&quot; Some prestigious universities like Caltech, Stanford, MIT and Berkeley never disclose their information regarding their cut off major and cumulative GPA for M.Sc and PhD programs in engineering. I also aware of the fact that graduate schools will look at the whole package and make their decision based on sets of factors like letter of recommendation, statement of purpose as well as GPA.   However, since application fee is pretty high it is not worth it to apply to schools that you know you have a slight chance of getting into it so, I would like to know what would be the acceptable GPA (Both major and Cumulative) for applicants to be competitive for top five schools in the world? I want to return two objects from a Java method and was wondering what could be a good way of doing so?  The possible ways I can think of are: return a HashMap (since the two Objects are related) or return an ArrayList of Object objects.  To be more precise, the two objects I want to return are (a) List of objects and (b) comma separated names of the same.  I want to return these two Objects from one method because I dont want to iterate through the list of objects to get the comma separated names (which I can do in the same loop in this method).  Somehow, returning a HashMap does not look a very elegant way of doing so. In the answer , the parser breaks my URL link.  The URL was with __word__:      http://www.open-mpi.de/projects/hwloc/doc/v1.2/group__hwlocality__membinding.php#gade5e2c28ea8475a479bf2b1df36c6ccd   After parsing, it becomes (&lt;a href is broken; no clickable link):  hwlocality_membinding.php#gade5e2c28ea8475a479bf2b1df36c6ccd"&gt;http://www.open-mpi.de/projects/hwloc/doc/v1.2/group_hwlocality_membinding.php#gade5e2c28ea8475a479bf2b1df36c6ccd   Short test:  http://example.com/_test_ -> test"&gt;http://example.com/test  Or if I put in a naked URL with _test_ just here: Force figure placement in text I have a problem when a lot of figures are in question. Some figures tend to "fly around", that is, be a paragraph below, although I placed them before that paragraph. I use code:  \begin{figure}[ht] \begin{center} \advance\leftskip-3cm \advance\rightskip-3cm \includegraphics[keepaspectratio=true,scale=0.6]{slike/visina8} \caption{} \label{visina8} \end{center}\end{figure}   to place my figures. How can I tell latex I REALLY want the figure in that specific place, no matter how much whitespace will be left? What is the Python equivalent of static variables inside a function? What is the idiomatic Python equivalent of this C/C++ code?  void foo() {     static int counter = 0;     counter++;     printf("counter is %d\n", counter); }   specifically, how does one implement the static member at the function level, as opposed to the class level? And does placing the function into a class change anything? Is it true that a dihedral group is nonabelian? Is it true that a dihedral group is nonabelian?  I'm not sure if the result is true. I checked it for some lower order and I think the result may correct.   But I failed to prove/disprove the result.