Prove that for $n \geqslant 1,$ the integer $n(n+1)(2 n+1)$ is divisible by $6.$   My attempt. We enough to show that $2|n(n+1)(2 n+1)$ and $3|n(n+1)(2 n+1)$. Note that any integer $n$ is of the form: $2k$ or $2k+1.$  So $n(n+1)(2n+1)=2k(2k+2)(4k+1)$. Hence $2|2k(2k+2)$, that is $2|n(n+1)$.   How can I continue this proof? Can you help?
I've been asked to show that:  $n(n+1)(2n+1) \equiv 0 \pmod 6$  I found in a previous question that:  $n(n+1)$ was divisible by $2$ and resulted in an even number e.g  $n(n+1) \equiv 0 \pmod 2$  so I figured I needed to find:  $(2n+1) \equiv 0 \pmod 3$ in order to complete  $n(n+1)(2n+1) \equiv 0 \pmod 6$  but I am unsure on how to find $(2n+1) \equiv 0 \pmod 3$  Is this the right way to find the mod 6 and if so could you tell me how I could find  $(2n+1) \equiv 0 \pmod 3?$
What is the physical significance of the imaginary part when plane waves are represented as $e^{i(kx-\omega t)}$? I've read that plane wave equations can be represented in various forms, like sine or cosine curves, etc. What is the part of the imaginary unit $i$ when plane waves are represented in the form $$f(x) = Ae^{i (kx - \omega t)},$$ using complex exponentials? How to place a .png texture with transparency over a material I am trying to create a low poly face with a face from a .png file. The issue is I already have material placed for hair and skin, as shown in the picture below.   I have created a .png file with a face which I would like to overlay over the skin material on the front of the head.    All examples I searched so far assume I also want the color of the material from the .png texture.  Update:  I added UVMapping, changed view to texture, made sure one face is on the eye but nothing is displayed..... see image... I don't know what is going on. How to copy symbolic links? Remove HTML tags from a String In how many different ways can you split $10$ people into two groups of $5$ people? In how many different ways can you split $10$ people into two groups with the same amount of people?  My attempt: Since the order in which you choose someone doesn't matter, I chose to calculate the combinations: $10 \choose 5$ What is the limit of can I use same functions with the different type of argument inputs Heartbleed: are services other than HTTPS affected? The OpenSSL 'heartbleed' vulnerability () affects webservers serving HTTPS. Other services also use OpenSSL. Are these services also vulnerable to heartbleed-like data leakage?  I'm thinking in particular of   sshd secure SMTP, IMAP etc -- dovecot, exim &amp; postfix VPN servers -- openvpn and friends   all of which, on my systems at least, are linked to the OpenSSL libraries. Warning: preg_replace(): Unknown modifier Apply a modifier as a shapekey into existing shapekey or merge shapekeys? I am making a shapekey of a model shrinkwrapped to another model. The shrinkmap modifier is too inaccurate to wrap to the model alone. So, my first step was to roughly hand position the geometry around of the first model around the second, then use shrinkwrap. The rough positioning was done as a shape key. I can then apply shrinkwrap, but its as a separate shapekey. The two do not work well together. I need them to be the same shapekey. How do I apply a modifier to an existing shapekey, or merge/combine/join the two shapekeys? When I have my external monitor hooked up, I access it by mousing off the left hand side of my Macbook display. Can I switch this so that I can access the external by mousing off to the right of My Macbook display?   The reason I ask, is my monitor is on the right side of my macbook, and it's unintuitive to mouse left to get to a screen on the right. Strange floating-point behaviour in a Java program In my program I have one array with 25 double values 0.04 When I try to sum these values in a loop I get following results:  0.0 + 0.04 = 0.04 0.04 + 0.04 = 0.08 0.08 + 0.04 = 0.12 0.12 + 0.04 = 0.16 0.16 + 0.04 = 0.2 0.2 + 0.04 = 0.24000000000000002 0.24000000000000002 + 0.04 = 0.28 0.28 + 0.04 = 0.32 0.32 + 0.04 = 0.36 0.36 + 0.04 = 0.39999999999999997 0.39999999999999997 + 0.04 = 0.43999999999999995 0.43999999999999995 + 0.04 = 0.4799999999999999 0.4799999999999999 + 0.04 = 0.5199999999999999 0.5199999999999999 + 0.04 = 0.5599999999999999 0.5599999999999999 + 0.04 = 0.6 0.6 + 0.04 = 0.64 0.64 + 0.04 = 0.68 0.68 + 0.04 = 0.7200000000000001 0.7200000000000001 + 0.04 = 0.7600000000000001 0.7600000000000001 + 0.04 = 0.8000000000000002 0.8000000000000002 + 0.04 = 0.8400000000000002 0.8400000000000002 + 0.04 = 0.8800000000000002 0.8800000000000002 + 0.04 = 0.9200000000000003 0.9200000000000003 + 0.04 = 0.9600000000000003   Why on earth could that happen?! Unable to boot into Windows after installing Ubuntu, how to fix? I installed Windows on my computer, followed by an installation of Ubuntu. However, now I'm unable to boot into my Windows install.  What can I do to fix this? Why WordPress choose data serialization over json_encode? In my little age with WordPress, I've seen WordPress itself and its friendly plugins are using PHP serialize() in storing data into db in many cases. But in a recent search I found a serious community support for the json_encode() over the serialize().     And I personally tested an associative array with both of 'em, that shows:  serialize() stores 342 chars json_encode() stores 285 chars  Why I'm asking this? I'm on a project while I'm going to store repeating meta fields to a post. Where:  Data would be basically in English, but sometimes can be Bengali Data would be associative array, 3 level deep (I hope I understood levels correctly):   array(     1 =&gt; array(         'key'=&gt;'value',         'key2'=&gt;'value'     ),     2 =&gt; array(         'key'=&gt;'value',         'key2'=&gt;'value'     ) )  I've checked the postmeta table's meta_value field it's a longtext,  a length of 4,294,967,295 chars (4GB). So I need a robust solution into storing things. Proofs of Determinants of Block matrices Creating your own Tinyurl style uid intuition on the fundamental group of $S^1$ I am familiar with the proof that the fundamental group of the unit circle $S^1$ is $\mathbb Z$, yet I couldn't develop intuition for why it is true. For example, why would I fail if I try to find homotopy from a path that circles $S^1$ once to the path that stays at one point all the time? What brakes the continuity of such a homotopy? Pixel RGB with ImageMagick and Rails How to stop Skype from starting automatically when booting Windows 10? List sorting by custom field I dont know how to sort a list of SObject Case.  I have a List&lt;Case&gt; newCases which i iterate over in for cycle, where i save Cases to my new List&lt;Case&gt; casesList.  for (Case newCase : newCases) {             if(newCase.Case_Content__c == '111' &amp;&amp; newCase.Status == '24'){                 String cuid = newCase.CUID__c;                 cuidSet.add(cuid);                 casesList.add(newCase);                 casesList.sort(); //TODO sorting by CUID__c             }         }   What i need to do is, to sort this new list by field CUID__c. So the Cases with same CUID are together.  Thank you for any help :)