If $(a,b)$, $(a,y)$ and $(b,x)$ = $1$, show that $(ab, ax + by)$ = $1$ I tried to do this by showing that $(a,by)$=$1$ and $(b,ax)$=$1$. But then I got completely stuck. I tried to use Bezout's theorem, but ended up with nothing.
If $\gcd(a,b)=1$, $\gcd(a,y)=1$ and $\gcd(b,x)=1$ then prove that $ax+by$ is prime to $ab$.  I tried assuming Diophantine Equations for all the relations and representing everything in terms of $a$, but that didn't lead me anywhere. There must be an intuitive and elegant approach right?
$\sum a_n$ be convergent but not absolutely convergent, $\sum_{n=1}^{\infty} a_n=0$ Let $\sum a_n$ be convergent but not absolutely convergent, $\sum_{n=1}^{\infty} a_n=0$,$s_k$ denotes the partial sum then could anyone tell me which of the following is/are correct?  $1.$ $s_k=0$ for infinitely many $k$  $2$. $s_k&gt;0$ and $&lt;0$ for infnitely many $k$  3.$s_k&gt;0$ for all $k$  4.$s_k&gt;0$ for all but finitely many $k$  if we take $a_n=(-1)^n{1\over n}$ then $\sum a_n$ is convergent but not absolutely convergent,but I don't know $\sum_{n=1}^{\infty} a_n=0$? so I am puzzled could any one tell me how to proceed? $$\frac{1}{n+1}&lt; \log(1+ 1/n)$$  Any ideas? I tried estimating the difference between $1/n$ and the logarithm and comparing with $1/n-1/(n+1)$ but I miserably failed. What's a good back-up strategy for 1 desktop PC? strange character replacement on SO I added a comment to  but something strange is happening. The end code reads "); but SO decided to change that to ";);.    I did not add a ; in front of the ).  To further explain what's happening here, the following:  "http://example.com/." http://example.com/. "http://example.com." http://example.com. "http://example.com" http://example.com   renders as: I'm a bit confused in describing 1.5hrs in words.   Is writing one and a half an hour correct or should it be one and half hour?  E.g:     I'll see you there in one and a half an hour.     OR       I'll see you there in one and half hour.   Or is there any other correct way of writing this? Can't cast to to unspecific nested type with generics How to calculate $\lim_{x\to 1^+} \log (x)^{\log(x)}$? How to calculate $\lim_{x\to 1^+} \log (x)^{\log(x)}$ ? i know that its "1", but why? How can i calculate this?  Thank you very very much =) Anthem-esque short story about a dude who is very smart and strong so he wears devices to slow him and weaken him I remember reading a short story back in high school that was similar to Anthem by Ayn Rand about this guy who was very strong and smart. And because everyone in this world had to be equal he was forced to wear weights to slow him down, and he had to wear some device on his head that made it hard for him to see and hear, breaking his focus.  At the end of the story in breaks into a news station and tries like broadcasting a message. And he runs away with one of the ballerinas who has blocks on her feet so it makes it hard for her to dance. What does the '!!!' syntax mean in javascript? What is the string concatenation operator in Oracle? What is the string concatenation operator in Oracle SQL?    Are there any "interesting" features I should be careful of?    (This seems obvious, but I couldn't find a previous question asking it). Is it possible to export a single object from an entire scene? Sharepoint 2010 - Deploy webpart as Feature scope = web Limit of $x\left(\left(1 + \frac{1}{x}\right)^x - e\right)$ when $x\to\infty$ I am stuck on how to calculate the following limit: $$\lim_{x\to\infty}x\left(\left(1 + \frac{1}{x}\right)^x - e\right).$$  Definitely, it has to be through l'Hôpital's rule. We know that $\lim_{x\to\infty} (1+(1/x))^x = e$. So, I wrote the above expression as $$\lim_{x\to\infty}\frac{\left(1 + \frac{1}{x}\right)^x - e}{1/x}.$$Both numerator and denominator tend to zero as x tends to infinity. I applied l'Hôpital's rule twice, but I got the limit equal to infinity which is clearly wrong. In the book, it says that the limit should be $-e/2$.  Any help please? Also, in case someone managed to solve it, please tell me which numerator and denominator did you apply your l'Hôpital's rule to in both cases? Thanks a lot  Also, guys can you tell me how to write equations in this forum? Thanks. I have to prove this : "$n^7-n$ is a multiple of 7". This is what I have done this so far :  $P(n):n^7-n$ On putting $n=1,$ $P(1):1^7-1=0$, which is a multiple of 7. So, $P(1)$ is true. Let $P(k)$ be a multiple of 7. So, $k^7-k$ is a multiple of 7. So, $k^7-k=7m$, where $m$ is any natural number. Let $P(k+1)$ be true, Hence, $(k+1)^7-(k+1)$ is a multiple of 7.  Now, how do I evaluate $(k+1)^7$ without using the binomial theorem ? Or is there some other way to do it ? How to select distinct field values using Solr? Perform feature normalization before or within model validation? Showing that $\sum_{i=1}^n \frac{1}{i} \geq \log{n}$ Linear algebra is a very well-behaved part of mathematics. Soon after you have mastered the basics you got a good feeling for what kind of statements should be true -- even if you are not familiar with all major results and counterexamples.  If one replaces the underlying field by a ring, and therefore looks at modules, things become more tricky. Many pathologies occur that one maybe would not expect coming from linear algebra.  I am looking for a list of such pathologies where modules behave differently than vector spaces. This list should not only be a list of statements but all phenomena should be illustrated by an example.  To start the list I will post an answer below with all pathologies that I know from the top of my head. This should also explain better what kind of list I have in mind. How to efficiently count the number of keys/properties of an object in JavaScript? What's the fastest way to count the number of keys/properties of an object?  It it possible to do this without iterating over the object?  i.e. without doing  var count = 0; for (k in myobj) if (myobj.hasOwnProperty(k)) count++;   (Firefox did provide a magic __count__ property, but this was removed somewhere around version 4.) Why use hex? Hey! I was looking at this code at   I noticed that in some situations they used hex numbers, like in line 134:  for (j = 1; val &amp;&amp; j &lt;= 0x80; j &lt;&lt;= 1, q++)   Now why would they use the 0x80? I am not that good with hex but I found an online hex to decimal and it gave me 128 for 0x80.  Also before line 134, on line 114 they have this:  small_n = (n &amp; 0xffff0000) == 0;   The hex to decimal gave me 4294901760 for that hex number. So here in this line they are making a bit AND and comparing the result to 0??  Why not just use the number? Can anyone please explain and please do give examples of other situations.  Also I have seen large lines of code where it's just hex numbers and never really understood why :(