Does the equation $x^3+10x^2-100x+1729=0$ fas atleast one complex root $\alpha$ such that $|\alpha|$ $\gt 12$??  Since any cubic equation has atleast one real root, let it be say $k$. Let the other two complex roots be $\alpha+i\beta , \alpha-i\beta$. Then The product of the roots is $(\alpha^2+\beta^2)k=-1729.$ Since $(\alpha^2+\beta^2) \ge 0$, $k\lt0$.   From here I am unable to find any roots or deduce any further.
How do I show that the equation $x^3+10x^2-100x+1729$$=0$ has at least one complex root $a$ such that $|a|$$&gt;$$12$.
Execute script after specific delay using JavaScript Is there any JavaScript method similar to the jQuery delay() or wait() (to delay the execution of a script for a specific amount of time)? How do I display the nth result of an ls command? I used this command to display the first result of files in my directory.    ls | head -n 1   My simple question is, how can I modify this command to display say the nth result?  Thanks! Every integer is congruent to the sum of its digits mod 9 This is a question from the free Harvard online abstract algebra .  I'm posting my solutions here to get some feedback on them.  For a fuller explanation, see  post.  This problem is from assignment 6.  The notes from this lecture can be found .     Prove that every integer $a$ is congruent to the sum of its decimal digits modulo 9.   Let $s$ be an $(n+1)$-digit integer given by $s=a_n\cdot10^n+\dots+a_2\cdot10^2+a_1\cdot10^1+a_0\cdot10^0$ with $a_i\in\mathbb{Z}$ such that $0\leq a_i\leq9$.  Then $ \begin{align} s\,\mathrm{mod}\,9&amp;=\bar{s}\\ &amp;=\overline{a_n\cdot10^n+\cdots+a_2\cdot10^2+a_1\cdot10^1+a_0\cdot10^0}\\ &amp;=\overline{a_n\cdot10^n}+\overline{\cdots}+\overline{a_2\cdot10^2}+\overline{a_1\cdot10^1}+\overline{a_0\cdot10^0}\\ &amp;=\overline{\overline{a_n}\cdot\overline{10^n}}+\overline{\cdots}+\overline{\overline{a_2}\cdot\overline{10^2}}+\overline{\overline{a_1}\cdot\overline{10^1}}+\overline{\overline{a_0}\cdot\overline{10^0}}\\ &amp;=\overline{\overline{a_n}\cdot1}+\overline{\cdots}+\overline{\overline{a_2}\cdot1}+\overline{\overline{a_1}\cdot1}+\overline{\overline{a_0}\cdot1}\\ &amp;=\overline{a_n}+\overline{\cdots}+\overline{a_2}+\overline{a_1}+\overline{a_0}\\ &amp;=a_n\,\mathrm{mod}\,9+\cdots+a_2\,\mathrm{mod}\,9+a_1\,\mathrm{mod}\,9+a_0\,\mathrm{mod}\,9 \end{align} $  Again, I welcome any critique of my reasoning and/or my style as well as alternative solutions to the problem.  Thanks. $a\equiv\bar a\!\pmod{\!kn}\Rightarrow a\equiv\bar a\!\pmod{\! n};\ $ $(a\bmod kn)\bmod n=a\bmod n.\ $ Congruences persist mod factors of the modulus I noticed relation between modulo operation and number which is power of two  Example I have to calculate $ 3431242341 \mod 2^5 $, which is $ 5 $ but it is equivalent to  $ ( 3431242341 \mod 2^9 ) \mod 2^5 $   I tried many examples and it seems to be true in general, and I am not sure if it is a coincidence or true in general that I can use first modulo operation ( greater number) and the result will be the same. How do you backup all your emails, by copying the 'Outlook' folder? (Outlook 2013) [...] All versions of Outlook use pst files, but the disk location has changed over the years. Right-click your email account on the left in Outlook and select Data File Properties. Click Advanced to see the filename. Only the pst file is essential, but copy the whole folder to back up Outlook’s settings and preferences too. Just copy it back to restore it, or click File, Open, Open Outlook Data File. Outlook can work with any pst file stored anywhere.  Backing up your Outlook data file(s) is as easy as copying your .OST or .PST files over to another hard drive, cloud server, thumb drive, or some other storage medium. You remember how to locate your data files? Open the Mail control panel and click the “Data Files” button. Click “Open File Location…” and you can then back up your data files. [I don't display  here)  After following the instructions above, I arrive at:  C:\Users\[my username]\AppData\Local\Microsoft\Outlook  However, all Outlook Data Files here are .ost, and NOT .pst. In  File &gt; Open &amp; Export &gt; Open Outlook Data File, 'Open Outlook Data File' only works for .pst. What is wrong here? Where's the folder with all the .pst files? How can I backup all my Outlook email accounts simultaneously? How can I completely remove all traces of Java on my system?   I already know  in case I need it again. Can a child enter the UK on an accompanied visa and then leave the country alone? My daughter who is just under 18 has a child accompanied visa to the UK. I am traveling with her to the UK for a few days. We return from UK on the same day except that she returns to India and I am heading to San Francisco an hour later on the same day.   Is that okay or will we have an issue at immigration into the United Kingdom? Independence of disjoint events Find lost device without Gmail account configured and without SIM card I recently misplaced a phone that did not have gmail account configured and did not have a SIM card in it either. Is there any way I can track it ?   All of the tracking apps either require sending an SMS (needs SIM card) or OTA install(needs Gmail account).   I do have the IMEI number of the phone. Recently the cranks on my bike started to creak. I checked with my usual "bike-bible" . They recommended dis-assemble, lubricate, re-assemble.  So I did that.  Remove cranks. Remove spindle. Clean. Lubricate. Reassemble.  No, the creak is still there.  (To find out the source-area of the sound I had a friend riding the bike slowly while I ran alongside listening. And the sound where coming from the area of the bottom bracket/cranks. I am happy no neighbor saw me...)   For each revelation of the cranks there are two creak-noises.  If I, when on the bike, push down with both feet I hear a noise. Turn 180 degrees without pressure, push down again and there is the noise again. If I am "too soft", low pressure there are no sound. Since I have clips I have tried to only work with one foot to keep a constant load during a revelation, there where no detectable sound then.  Anyone have any idea what is wrong on the bike? And how I should fix it?  /Leif How common is "thrice"? Having trouble adding objects to NSMutableArray in Objective C "Fatal error: Cannot redeclare " Is every solution of Einstein field equations unique? Einstein's equation is  $$8 \pi T_{ab} = G_{ab},$$  where the left side contains the stress-energy tensor and the right side contains the Einstein tensor.  Is there exactly one unique stress-energy tensor corresponding to a given spacetime curvature? Or is it possible for one curvature (i.e. one spacetime metric) to be produced by multiple different mass configurations? I have a wireless Access point which is connected to internet and the same being accessing by 2 system, one on windows XP and the other on Ubuntu.  Please let me know if it is possible to access the c:\ d:\ or any drives on windows xp machine directly from ubuntu system ?  I have tried doing this with 2 windows xp system but not sure how to do this from ubuntu.  Any help regarding the same is highly appreciated. (reason to do so is to access files in windows xp machine)  -Thanks &amp; regards, Manju Help finding a combinatorial proof of $k {n \choose k } = n {n - 1 \choose k -1}$ Help finding a combinatorial proof of $k {n \choose k } = n {n - 1 \choose k -1}$  I have expanded it this far:  $$\frac{k \cdot n!}{k!(n-k)!} = \frac{n \cdot (n-1)!}{(k-1)!(n-k)!} $$  but then I am stuck from where to go from there. Give me a blank Comment box when I am Improving a Suggested Edit The suggested edit screen includes both the suggested edit, and the reason for the edit:        When you click to improve the edit, that comment is left as-is and hidden at the bottom of the post above the save edit button:        At the moment, when I Improve a Suggested Edit, the Comment text for that Edit is retained (though editable). It's unlikely that the comment would still apply (e.g. Suggester added a tag and Approver corrected a typo), so I would rather have the field blank.  Also, it's better to have a blank Comment than an incorrect one... many people don't bother changing the original Comment.  So that box should be blank like this instead: Assume $x$ doesn't occur free in $\alpha$, show that: $$\vdash (\forall x \beta \to \alpha) \leftrightarrow \exists x (\beta \to \alpha)$$   This is an exercise on page 130, A Mathematical Introduction to Logic, Herbert B. Enderton（2ed).   Here's my attempt: For the '$\to$' direction, since $x$ doesn't occur free in $\alpha$, we have $α→∀x α$ and $\forall x \alpha \to \alpha$ as logical axiom(In the second formula, we have$\alpha_x^x = \alpha$). Thus it suffice to show $\vdash (\forall x \beta \to \forall x \alpha) \rightarrow \exists x (\beta \to \alpha)$. By logical axiom $∀ x(α→β)→( ∀ x α→∀x β)$, it would suffice to show $\vdash \forall x (\beta \to   \alpha) \rightarrow (\exists x (\beta \to \alpha))$. It has been shown in a previous exercise that there's a deduction $∀ x ϕ→∃x ϕ$, so we are done for the "$\to$" part.  The problem for me is how to show the "$\leftarrow$" direction.  Added: Here's logical axioms that we can employ:         Tautologies(in the sense of propositional logic);   $∀ x α →α^x_ t $, where $t$ is substitutable for $x$ in $α$ ($α^x_ t$ is the formula derived from $\alpha$ by replacing $x$ by a term $t$);   $∀ x(α→β)→( ∀ x α→∀x β)$;   $α→∀x α$, where $x$ does not occur free in $α$.      Besides, generalization theorem and deduction principle can be utilized:     If $\Gamma \vdash ϕ$ and $x$ do not occur free in any   formula in$\Gamma$, then$\Gamma \vdash ∀ x ϕ$.      If $\Gamma ; γ \vdash ϕ,$ then $\Gamma \vdash  γ \to ϕ$. Suppose $G$ is a semigroup and it holds both left and right cancellation. Suppose $G$ is a semigroup and it holds both left and right cancellation. Also for each $a,b\in G$, $xa=b$ has solution in $G$. Prove G is a group.   I know this question looks very "old" style. First I think it's easy for me too since I already practice probelems like "semigroup, $xa=b$ and $ax=b$ have solution , then group" Or "finite semigroup, both left and right cancellation hold, then group ". But I still get stuck... My try:  First of course, for $xa=a$ , there exists $e_a$ such that $e_aa=a$. Now we need to proof $e_a$ is left identity for all elements. Then left inverse exists hence $G$ is a group. But the key is to proof $e_ab=b$ for all $b\in G$. How to find out the default file manager in the terminal?