How can I prove that if the cube of a number is a multiple of 2, then that number is a multiple of 2
Theorem:   If $x$ is a positive integer and  $x^3$ is even, then $x$ is even.   My Proof by Contrapositive:  I. Assuming that $x$ is odd, then I will show that $x^3$ is odd.  II. $x$ is odd, so $x$ can be expressed as $2k+1$  III. $(2k+1)^3$  IV. $2k(4k^2 + 6k + 3) +1$  V. Since $2k(4k^2 + 6k + 3) + 1$ is an integer, we can conclude $x^3$ is odd.  Did I go about this correctly? Should step V say "$2$ multiplied by an integer and added by $1$ is odd therefore we can conclude $x^3$ is odd"?
$a_1 a_1', a_2 a_2', ..., a_n a_n'$ is not a reduced modulo system $\mod p$. If $a_1, a_2, ..., a_n$ is a reduced residue system $\mod p$ and $a_1', a_2', ..., a_n'$ is another reduced residue system $\mod p$ then $a_1 a_1', a_2 a_2', ..., a_n a_n'$  is not a reduced modulo system $\mod p$.  Not able to show a counter to it. Help Needed. Is there any way to get the ID of the element that fires an event?  I'm thinking something like:    $(document).ready(function() {   $("a").click(function() {     var test = caller.id;     alert(test.val());   }); }); &lt;script type="text/javascript" src="starterkit/jquery.js"&gt;&lt;/script&gt;  &lt;form class="item" id="aaa"&gt;   &lt;input class="title"&gt;&lt;/input&gt; &lt;/form&gt; &lt;form class="item" id="bbb"&gt;   &lt;input class="title"&gt;&lt;/input&gt; &lt;/form&gt;     Except of course that the var test should contain the id "aaa", if the event is fired from the first form, and "bbb", if the event is fired from the second form. Homology Groups of $S^n$ Aufbau principle in modern quantum theory Where to find structure of objects that I can access in PHP snippets added to different parts of UI? elementary row operation change the eigenvalue of a matrix I know that elementary row operations may change the eigenvalues of a matrix, but if these operations don't change the determinant, why should change the eigenvalues??  Thanks format date in c# How can I format a date as dd/mm/yyyy or mm/dd/yy ?   Like in VB format("dd/mm/yy",now)  How can I do this in C#? How to prepare files for rsync on a case insensitive filesystem? Install java Runtime Environment How do I calculate number of hours > 24 in Excel? I've got an excerpt of my spreadsheet below. It's a time tracking application containing a start date/day, a session ID and the duration of the session. Column A is populated by adding the duration in Column D. I'm trying to total the durations, but keep getting incorrect values in cases where the total duration is > 24 hours.  Consider        A                   B        C                D          1  2012/05/22 01:00  |  Wed  |  Session 23341  |  9:00 2  2012/05/23 10:00  |  Wed  |  Session 23342  | 22:00 3  2012/05/25 08:00  |  Fri  |                 |  0:00   As you can see, the date differential between A2 and A3 demonstrates that the value of 22:00 in D2 represents a duration of 46 hours (05/23 10:00 + 46:00 = 05/25 08:00). However, Column D is only displaying the 22:00 in excess of a full 24 hour day.  My attempts to read D2 have included int(d2) = 0.00  /  int(d2)*24 = 0.00  /  hour(d2) = 22.00  /  int(a3-a2)*24 =0.00 etc. and have been unsuccessful in pulling a value of "46".   How can I accurately represent D2 to achieve accurate summary data? Under TeX, when I write a line too long, it prints a fat black box at the end of the line.  That is probably a feature, but I dislike it.  How to turn that OFF? Is there a command-line option or something else?   I usually use LaTeX, but I have a complicated file which happens to be plain TeX. Design Review: 3.3V Logic to 12V Logic (Uni-directional and Bi-directional) I'm looking for feedback and improvements on my design below. Specific questions are at the end. I created this as an attempt to answer my own, . From my feedback there, it seems like I needed to be more clear in my requirements. These are screenshots from Eagle. If there's a better way to share, please let me know so I can correct it. Entire board  The overall purpose of this board is to translate between 12V logic, and 3.3V logic. The top portion of the board represents the 12V side. The LEDs are placeholders to indicate whether the 12V signal is HIGH or LOW. In practice, these would be replaced with more , requiring up to 3A. The switches are also placeholders, allowing one to manually turn the 12V signal on and off. They are only for test purposes, the &quot;real&quot; 12V signal would be coming as the output from other relays. The bottom half is the 3.3V side. This will be connected to a  (BBB). The board will connect to 2 output GPIO pins and 2 input GPIO pins of the BBB. Overall, I want to translate the 3.3V I/O from the BBB to the 12V signals on the other side (to control more relays and read their signals). I have designed 3 modules to control both directions individually, and together. 3V3_TO_12V Module  This module converts the 3.3V GPIO output from the BBB to a 12V signal on the other side. When the 3.3V signal is low, the 12V signal is low. When the 3.3V signal is high, the 12V signal is high. To do this, the 3.3V GPIO output is connect to a , which turns a  relay on and off. This switches the 12V power as output of the module to be used elsewhere. 12V_TO_3V3 Module  This module converts a 12V signal to 3.3V signal, to be read by a GPIO input from the BBB on the other side. When the 12V signal is low, the 3.3V signal is low. When the 12V signal is high, the 3.3V signal is high. To do this, the 12V signal simply turns a  relay on and off. This switches the 3.3V power to be read as input by the BBB. DUAL_12V↔3V3 Module  This module combines the previous two modules to create a bidirectional translation. If the 12V side is high, the 3.3V input to the BBB GPIO should be high. If the 3.3V output from the BBB GPIO is high, then the 12V side should be high too. It's similar to a logical OR, if either side is high, the corresponding signal on the other side should be high too. A single 12V signal has to be split into individual input and output signals on the 3.3V side for the BBB. Questions  Is the  and  relay combination overkill for having the BBB GPIO output control the 12V power line? I like the isolation and safety it provides, but it seems too much (in terms of cost and board space) Is there a better solution for the 12V to 3.3V conversion. I like the simplicity of the  relay switching a 3.3V signal, but it takes up lots of room. Does the DUAL_12V↔3V3 Module (which combines each translation direction) even make sense? This seems wasteful to use 2 relays and a high side driver, but I haven't found simpler solution that achieves the same result. Should I add pulldown resistors for the BBB GPIO inputs, or are the internal pulldown resistors good enough?  Any other feedback is welcome. I'm new to electronic design. When I tried to run , I got the below screen:    The captcha is missing!  I tried in other browsers too.    Update:  From @Oded's comments, I changed to HTTP and it was working perfectly.  Out of curiosity I would like to know why the captcha isn't working when accessing Data Explorer over HTTPS. To be specific, I was trying this code:  package hello;  public class Hello {      Clock clock = new Clock();      public static void main(String args[]) {         clock.sayTime();     } }   But it gave the error      Cannot access non-static field in static method main   So I changed the declaration of clock to this:  static Clock clock = new Clock();   And it worked. What does it mean to put that keyword before the declaration? What exactly will it do and/or restrict in terms of what can be done to that object? Harmonic Motion How to automatically hyphenate within \texttt? This question was phrased as a bug report for LaTeX and I thought it would be helpful to be presented here (including an answer).  The person had a lot of "common words" in typewriter font sprinkled around his text and he was surprised that this text would not automatically hyphenate and he had to manually insert \- to mark hypenation points. There are numerous (, , ) questions concerning this problem. I realize this is a borderline duplicate, but I'll re-iterate the futility of this particular filter and why it should be outright removed from Stack Overflow.  Do you know how effective this is in preventing users who just don't care from posting questions with bad titles? 100% ineffective. What do these users do?   Replace it with a synonym: "issue," "trouble," "predicament," "botheration" or what have you. Or worse, they simply work around the filter by deliberately misspelling the word or inserting space or punctuation:    () ?    and the coup de grâce:      I mean, even Jeff . This adds zero value to the title. In fact, it looks worse. Why, oh why are people  to scrap a bad feature? All this does is:   Prevent people who have a  to use the word "problem" in the title from posting questions with an appropriate title. Break the search feature because of workarounds or the need to use less descriptive titles than the error itself   This is an extraordinarily useless (and harmful) feature. Please, for the love of Cthulhu,  remove, eliminate, abolish, eradicate, extirpate this "feature" from Stack Overflow.    Bonus:     So I can't mention the Halting Problem in the title? – Keith Thompson Nov 1 at 19:05      @KeithThompson Maybe it's a "Halting Issue". Or "Need help with Halting". Or the "Halting thing in Computability Theory". – slhck Nov 1 at 19:14 Finding the nth non square free number We all know what  numbers are. I was more interested in checking an efficient way to find nth non square free number. Why can't I change the permissions of files I have access to? I designed a poster in CorelDraw X5 for printing purposes. I was told by the Printshop guy to use colors only in CMYK. For the most part I did, I don't know why I missed it but I used one color in RGB. I liked this color so much that I've used it at many places. So, current situation is that some colors are in RGB and some are in CMYK. So, I want to convert all RGB to CMYK and thought the problem would be solved. But now I'm actually facing a lot of trouble. The colors are not at all same. I hate the colors that it is showing me.  This is the color I used in RGB:    When I changed the "Model" to CMYK, the color it shows is really different. As you can see in the picture below, it's too dull:    The RGB values of both colors are not the same. What can I do to get the same color in CMYK? I did a little bit of reading here and there. They say, that you can't get same. In that case, what is the closest color that I can get?