If $a^6 \mid b^6$, where $a,b\in \mathbb{Z^{+}}$, then does $a\mid b$?    I was thinking of using the Fundamental Theorem of Arithmetic to show that $$a^6=\prod_{i}p_{i}^{6k_i}\quad\text{and}\quad b^6=\prod_{i}p_{i}^{6q_i}$$ And, since $6k_i\leq 6q_i$, $\forall i$, $k_i \leq q_i$ which implies $a\mid b$.   Is this correct?
I would like to show that $a^n \mid b^n$ implies $a \mid b$  I thought I could convert it to congruences and work backwards, but as far as I remember, $a \equiv b \pmod{m}$ implies $a^n \equiv b^n \pmod{m}$, not the opposite, unless $m$ is prime. Is that right?  So I am not sure how to approach this one. Any ideas?  Thanks!
matlab based program where output comprises of sum of rows of input It looks like this guy  without even knowing it.  I don't get the impression that he was trying to harm anyone with this process.  What about a warning message that triggers after a person upvotes (or downvotes) the same person X times within Y time span (with X and Y being thresholds that are well below the limit that actually results in a vote fraud reversal)?  Say, twice within 15 minutes?  Maybe display a warning pop-up that says:     Whoa, Camel!  It looks like you're starting to perform serial voting on a particular user.  Did you know that    I have little doubt that this guy would have stopped had he been (gently) warned.  Perhaps others will stop, too. Electrodynamics textbooks that emphasize applications Please recommend undergraduate-level textbooks on electrodynamics which emphasize practical applications and real life examples. Please describe the book's level and contents and its intended audience in as much detail as possible.   Please provide both applications from everyday life for curious students as well as engineering applications from a physics perspective. Erasing a Linux laptop I have a laptop with Debian on it, and I am going to sell this laptop.   Would it suffice to erase the Debian installation before selling it to completely clean up my laptop from my personal data, and if yes how can I uninstall Debian (so that there isn't any operating system on the laptop)? What is the difference between a synchronized method and synchronized block in Java? I am trying get an unused server up and running, it booted fine but i was greeted with the login window and none of the known passwords seem to work. The server I am trying to get to work is not a Domain Controller and has not been used in a long time.  OS: Windows 2003 Server R2 Enterprise Edition  I tried Ophcrack XP (there is no WinServer version of Ophcrack) - it couldn't locate the partitions with hashes. I tried mounting the partitions myself (#mount /dev/hda*x* /mnt/ntfs), but it gave me an error saying /dev/hda*x* didn't exist.  Next I tried booting into a Ubuntu 10.04.3 LiveCD so that I could install and use chntpw - no go, Ubuntu Live wouldn't boot.  Based on what I've read up the following tools may be of help:   Windows Password recovery Petter Nordahl-Hagen's Offline NT Password &amp; Registry Editor Openwall's John the Ripper EBCD – Emergency Boot CD - Bootable CD (R)ecovery (I)s (P)ossible Linux rescue system   I am hoping that someone here has prior experience with Win2k3 Administrator password resets so that I don't end up having to try all of the above to find out which one works. When is the graph of the divisor relation on $[n]$ planar? Mapping/texturing inside of a box I'm trying to replace the content of a box in a video, I have used the tracking system to match the movement of the video box with the interior of the one rendered in Blender, but I am running trough what seems a simple thing but I can't manage to get done.   I want the outside faces of the box to be invisible while hiding the inside of the box where the front wall of the box hides it, and the inside of the box to be rendered where visible. To do that I thought it would be as simple as making the face normals point to the inside of the box and giving a color only to the faces where normals are pointing to. But I can't manage to get that, am I going the wrong road? How can I achieve my goal? Given a dictionary, how can I find out if a given key in that dictionary has already been set to a non-None value?  I.e., I want to do this:  my_dict = {}  if (my_dict[key] != None):   my_dict[key] = 1 else:   my_dict[key] += 1   I.e., I want to increment the value if there's already one there, or set it to 1 otherwise. Is it possible to make the -init method private in Objective-C? I need to hide (make private) the -init method of my class in Objective-C.  How can I do that? What happens to folks who get question banned, by and large? There is a steady stream of duplicates of  coming in lately. It's got me wondering. Roughly speaking, out of say 100 folks who get this ban, how many    wander away and never interact with SO again?   do the sorts of things that might lift a ban, but don't get it lifted? do the sorts of things they should, and lift the ban? post a question of their own on Meta which we close as a dupe?   Among the continue-to-interact folks, is there a difference in success rate between those who post on Meta about it (motivated, but not good rules-readers, who nonetheless get customized advice) and those who do not (better rules readers, but working to some extent blind) - which do better?  Is there a reasonable chance of recovering from a ban, assuming you care enough to try? In vanilla Gmail, clicking on a conversation alters the fragment ID in the URL, allowing you to link back to that conversation.  (It opens in its own Gmail interface.)  I am using the preview pane, which is great, except that the URL no longer updates when viewing specific emails, so there appears to be no way to get the link to the currently-viewed conversation.  (The 'open in new window' button doesn't help, since it pulls its contents in on the client side; it isn't a standalone link.)  The only solution I've found is to turn off the preview pane, find the message again in the list, open it, grab the link, go back to the message list, and turn preview pane back on.  That's fairly onerous for something I do often though (to clip Gmail conversations to Evernote, where my GTD setup lives). Why does work depend on distance? What will at actually execute? Recovering lost photos on iPhone 5s after restoring from iTunes backup I used to have an iPhone 4s and all the data from there is synced in my iTunes in my Mac. I just got an iPhone 5s yesterday and decided then to plug it to my iTunes. There was a welcome message and 2 options: whether to set up as new iPhone or set up as old iPhone (4s contents).  So to be able to make everything fast, I chose the latter, but then all the new photos in my iPhone 5s were deleted and were replaced with all the contents from my 4s. Is there any way I could retrieve it? I used to work as a consultant and have been spoiled with nice looking slides. For instance, being able to create slides that have 2, 3, or 4 graphs (perfectly positioned and with nice headers!) at the click of a mouse button via macros.  Similarly, there are many other kinds of word slides that had multiple columns, etc. that would be nice to recreate as well as objects like "main point" boxes at the bottom that summarize a main point and ways to add a transparent highlight to any part of the screen.  Is this kind of functionality far beyond anything that LaTeX presentations can do or is there an archive somewhere of code for making more intricate presentations? How do I connect the initial highway to my six-lane road? I believe that the set of all $\mathbb{R\to R}$ continuous functions is $\mathfrak c$, the . However, I read in the book &quot;Metric spaces&quot; by Ó Searcóid that set of all $[0, 1]\to\mathbb{R}$ continuous functions is greater than $\mathfrak c$:  &quot;It is demonstrated in many textbooks that $\mathbb{Q}$ is countable, that $\mathbb{R}$ is uncountable, that every non-degenerate interval is uncountable, that the collection of continuous functions deﬁned on  $[0,1]$ is of a greater cardinality than $\mathbb{R}$, and that there are sets of greater and greater cardinality.&quot;  I understand that (via composition with the continuous function $\tan$ or $\arctan$) these sets of continuous functions have the same cardinality. Therefore, which claim is correct, and how do I prove this? I have just created a Minecraft 1.13.1 server and was wondering if there was any way to give everyone the teleport command even if they are not op. If not, is there a way that I could make them op but only have access to the teleport command? Where can I find PSD file or tutorial to create the effect of frame with shadow (see red arrow in image)