This problem is meant to be proved using contradiction and I've admittedly been stuck on it for some time. I'd really appreciate a point in the right direction, because it honestly feels like I've been getting nowhere.  This is all I've done:  Assume $x$ denotes the set of nine consecutive integers, and $x_1$ and $x_2$ denote the two partitions. We would have: $$\prod_{a\ \in \  x_1} a= \prod_{a\ \in \  x_2} a$$ $\implies$ $\prod_{a\ \in \  x_1} a$ and $ \prod_{a\ \in \  x_2} a$ are both even numbers, since both sets $x_1$ and $x_2$ would require to contain strictly odd numbers to have a product that is odd.  $\implies$ $x_1$ has at least two elements, one odd and one even.  $\implies$ $x_2$ has at most seven elements.  And that's where I hit a dead end. I thought perhaps I could represent the set $x$ as $$x := \{ (a_0, \ (a_0 +1), \ (a_0 +2), \ \dots , \ (a_0 +8) \}$$ where $a_0$ is any integer, then multiply things out and show in each case one side is less than the other, but then I noticed that'd be an extremely tedious route and that some of the more ugly expansions deserve some sort of proof by induction themselves to show whether or not they actually hold.  Please don't supply a proof that relies too heavily on a Number Theory-approach, as I don't have a background in that area therefore shouldn't rely on Theorems there to prove this.
Problem:   Prove that a set of nine consecutive integers cannot be partitioned into two subsets with same product   My attempt:  This problem would be solved if it could be proven that 9 consecutive numbers cannot exist such that all the numbers are of the form $2^a  3^b  5^c  7^d$ where $a,b,c,d$ are integers.  A suitable hint would be appreciated, I don not want a complete solution.  Edit:  If there existed a prime factor greater than $7$ of any number present in the set, then its multiple cannot exist b/c $22-11&gt;9$  Edit 2: As mentioned by Erick, my reasoning is not valid for numbers $1-9$ and $2-9$, let this be an exception.
How to ignore deprecation warnings in Python How to see the full date on macOS Big Sur? How to be granted root rights while browsing files/directories with Nautilus I'd like to know if there a way to be granted root rights while using Nautilus 2.30 ?   For example I'd like to move some old folders of long unused users from the home directory - remains of previous distros (Debian).   Of course I can open a terminal, but I want to know if it is possible to do that with a mouse in Nautilus. I've got this situation:  ./ ./myscript.sh ./arguments.txt ./test.sh   Inside myscript.sh, I have to run the file test.sh, passing to it the arguments contained inside arguments.txt.  myscript.sh is:  arguments=$(cat arguments.txt) source test.sh $arguments   This works well if if arguments.txt contains at most one argument:  firstargument    The substitution is:  ++ source test.sh 'firstargument'   But the problem is with two or more arguments. It does this:  ++ source test.sh 'firstargument secondargument'   Also, I don't know in advance the number of arguments inside arguments.txt. There can be zero or more. Forcing SSL and www in .htaccess I'm looking for a way to force SSL and www.  I've been able to force both separately but together I keep running into redirection issues. The following code works when handling a URL in this format: http://example.com and properly redirects to https://www.example.com but when the incoming URL is https://example.com it will not forward to https://www.example.com - Any suggestions?  EDIT: it should also send http://www.example.com to https://www.example.com.  RewriteCond %{REMOTE_ADDR} !127\.0\.0\.0 RewriteCond %{SERVER_PORT} 80 RewriteCond %{HTTP_HOST} !^www.example\.com$ RewriteRule ^(.*)$ https://www.example.com/$1 [R,L] This is problem 6.2 from the 3rd edition of Principles of Mathematical Analysis.      Problem 6.2: Suppose $f\geq 0$, f is continuous on $[a, b]$, and $\int_a^b f(x) \, dx = 0$. Prove that $f(x)=0$ for all $x \in [a, b]$.   I'm looking for a critique of my proof. It's a pretty easy problem, but I am always wary of making too bold of assumptions, especially on these low level/fundamental proofs. I'll be using Rudin's notation and refer to theorems from the text (If I should include the text of each theorem, feel free to leave a comment... I'm lazy but could probably use the TeX practice :p)  Proof: Assume, for contradiction, that $f&gt;0$. Then, for any partition $P$ we have the Lower Riemann Sum: $L(P, f)=\sum_{i=1}^n m_i \, \Delta x_i$. At least one $\Delta x_i$ must be positive, since $a &lt; b$, and each $m_i$ must be positive since we have $f&gt;0$ by assumption, so certainly $\sup f &gt; 0$. That means $L(P, f)&gt;0$. Thus, we have: $$0 = 0(b-a) &lt; L(P,f)\leq \sup L(P, f) = \inf U(P,f) =L$$ where the last string of equalities holds because our function is continuous on a compact interval, so is integrable by theorems 6.8 and 6.6. So our integral has value $L&gt;0$. This is in contradiction to our given assumption that $\int_a^b f(x) \, dx = 0$, so we must have that $f=0$ on $[a, b]$. $\Box$  So, I am wondering if my proof is correct (and is presented well). Also if someone could enlighten me as to what Rudin means when he say "Compare this with exercise 1," I'd be appreciative. Is it a hint or is there something else he expects you to notice? there are a lot of things I could compare :)...  Exercise 6.1: Suppose $\alpha$ increases on $[a, b]$, $a \leq x_0 \leq b$, $\alpha$ is continuous at $x_0$, $f(x_0)=1$, and $f(x)=0$ if $x\neq x_0$. Prove that $f$ is Riemann-Stieltjes Integrable and that $\int f \, d\alpha = 0$ Computing the Galois group of polynomials $x^n-a \in \mathbb{Q}[x]$ I have some problems with this exercise. I don't know if it can be done. Consider the polynomial $ x^n - a \in \mathbb{Q} $ Can I compute the Galois group of this over $\mathbb{Q}$? Maybe having a nice "basis.  The splitting field is given by $\mathbb{Q}(\zeta_n,\alpha)$ , where $\zeta_n$ is a primitive root of unity , and $\alpha$ is some number such that $\alpha^n = a $. Well first of all, I want a $\underline{"good basis"}$ for the splitting field. In the sense that the minimal polynomials, of the adjoined elements, are different (in this case the computation of the galois group is very simple).  For example one easy case, it's when $a&gt;0$, then $(a)^{\frac{1}{n}} \in \mathbb{R}$ , so clearly the minimal polynomial of $(a)^{\frac{1}{n}} , \zeta_n$ are distincs, and I'm done. If $n$ is odd then , it's also easy, since one root it's also real, for example $x^3-3 $, the real root is $ \root 3 \of { - 3}  =  - \root 3 \of 3  $ , so I can consider the splitting field as $\mathbb{Q}(-\root 3 \of 3 , \zeta_3 )=\mathbb{Q}(\root 3 \of 3 , \zeta_3 )$. The difficult case is when $n$ is even and $a&lt;0$ , for example  $x^8+20$ or $x^4+20$ in some cases as in the second, there are particular cases since there exist algorithms for the Galois group of quartics, but in general. It can be done?  $\underline{Remark}$ I'm searching a $\underline{"good basis"}$ for the splitting field. In the sense that the minimal polynomials, of the adjoined elements, are different since in this case the computation of the galois group is very simple. Can I update an early Airport Time Capsule? Hiding downvoted questions silences important discussions on Meta I just noticed a particular MSE post today on a topic I care about that was posted almost a week ago, which I entirely missed before. I'm a pretty avid MSE user, and this happens quite rarely to me.     was created by an SE employee, and is essentially the final answer to , which is the 10th most upvoted MSE post with a bit more than 900 upvotes right now. So it's kind of an important post, and on a topic I care about, but I still missed it entirely until it was linked in a comment somewhere else.  The reason I missed it is because it's heavily downvoted. This means that as it has below -8 score, it's hidden from the home page entirely, drastically reducing the visibility of the post. In particular, the negative score removes all visibility a question usually receives from being bumped due to edits, new answers and bounties.   Many of the recent heavily-downvoted Meta posts by SE have been featured manually, which grants them a lot of visibility beyond the particular meta site. This masked the problem pretty efficiently, but it is more of a workaround, and one that isn't applicable to all situations.  Voting means something different on meta sites, and not every downvoted question should be hidden from view. Closed downvoted questions can certainly still be affected by the usual rules on the home page, and that should cover the vast majority of questions that actually have a purpose being on the home page. But open questions should be displayed on the home page on meta sites even if they're heavily downvoted. Calculating a spherical polygon centroid Dimensionality of null space when Trace is Zero This is the fourth part of a four-part problem in Charles W. Curtis's book entitled Linear Algebra, An Introductory Approach (p. 216).  I've succeeded in proving the first three parts, but the most interesting part of the problem eludes me.  Part (a) requires the reader to prove that $\operatorname{Tr}{(AB)} = \operatorname{Tr}{(BA)}$, which I was able to show by writing out each side of the equation using sigma notation.  Part (b) asks the reader to use part (a) to show that similar matrices have the same trace.  If $A$ and $B$ are similar, then  $\operatorname{Tr}{(A)} = \operatorname{Tr}{(S^{-1}BS)}$  $= \operatorname{Tr}(BSS^{-1})$  $= \operatorname{Tr}(B)$,  which completes part (b).  Part (c) asks the reader to show that the vector subspace of matrices with trace equal to zero have dimension $n^2 - 1$.  Curtis provides the hint that the map from $M_n(F)$ to $F$ is a linear transformation.  From this, I used the theorem that $\dim T(V) + \dim n(T) = \dim V$ to obtain the dimension of the null space.  Part (d), however, I'm stuck on.  It asks the reader to show that subspace described in part (c) is generated by matrices of the form $AB - BA$, where $A$ and $B$ are arbitrary $n \times n$ matrices.  I tried to form a basis for the subspace, but wasn't really sure what it would look like since an $n \times n$ matrix has $n^2$ entries in it, but the basis would need $n^2 - 1$ matrixes.  I also tried to think of a linear transformation whose image would have the form of $AB - BA$, but this also didn't help me.  I'm kind of stuck...  Many thanks in advance! I'm trying to factorise $$ x^3z - x^3y - y^3z + yz^3 + xy^3 - xz^3 $$ into four linear factors.  By plugging it into WolframAlpha I've learned that it's $$-(x-y)(x-z)(y-z)(x+y+z)$$  My question is: what are the steps involved in factorising the expression? Is there a method I don't know about that I'd have access to with my limited maths?  Really appreciate any help! HTML code for an apostrophe Seemingly simple, but I cannot find anything relevant on the web.  What is the correct HTML code for an apostrophe? Is it &amp;#8217;? Right triangles and tangent lines Consider the hyperbola $xy=1$ with $x&gt;0$ and $y&gt;0.$ The tangent line at any point on the hyperbola forms a right triangle with the coordinate axes. Show that the area of the triangle does not depend on the point chosen. To show this, do the following:      a) Choose a point on the hyperbola of your choice. Find the tangent line and then find the area of the right triangle described above.      b) Choose a different point of your choice. Find the tangent line and then find the area of the right triangle described above.      c) Now for the general case, choose the point at $x=a.$ Find the tangent line at $x=a$ and then find the area of the right triangle described above.   This is the work I have so far:    No method to find the area of a right triangle was given, so I just used 1/2*B*H.  But I get stuck when given $x=a$ for part C.  Any ideas? Find number of files with a specific extension, in all subdirectories Is there a way to find the number of files of a specific type without having to loop through all results inn a Directory.GetFiles() or similar method? I am looking for something like this:  int ComponentCount = MagicFindFileCount(@"c:\windows\system32", "*.dll");   I know that I can make a recursive function to call Directory.GetFiles , but it would be much cleaner if I could do this without all the iterating.  EDIT: If it is not possible to do this without recursing and iterating yourself, what would be the best way to do it? Why use a dedicated light meter instead of the one built into the camera? In Adobe Photoshop I am able to select multiple layers at once with Shift+Click.  How can I do that in GIMP? Why does time change in Ubuntu after installing Windows Today I was installing Windows 7 on the same computer I have Ubuntu. For some reason, after installing Windows 7 and going back to Ubuntu the time went from the correct hour at that moment, 12:30PM to 7:30AM. This happened after installing Windows 7. For what I understand the BIOS has a time set, Ubuntu has the same time that is shown in the BIOS when installing it. At least up to this point is what I have seen. Now after installing Windows 7 (And believe me this is only for testing the performance of Windows 7 against Ubuntu in which case Windows 7 is falling behind a lot on the same hardware) the time changes in Ubuntu to the wrong one that is set in Windows 7.  I actually do not change the time in Windows 7 since this is one of the checks I am doing to see if the operating system can correctly grab the time. Windows 7 fails. Ubuntu does not BUT Windows 7 actually changes the time for Ubuntu if I boot with it afterwards.  So my question is, why/how does time change in Ubuntu after installing Windows.  NOTE - I have set Ubuntu time to "manual" instead of grabbing it from the Internet just to test out this. How do I change the default file manager back to Nautilus? This is something I just realised I don't entirely know the answer to, so I'm going to ask it here as I think it's interesting.  In most non-scientific writing on depth of field, diagrams generally show camera and subjects as being perfectly parallel, e.g.    However, is this a more accurate representation of the focal plane?    Are there ways to optically alter the shape of the focal plane?  Note: Obviously these diagrams are two dimensional, but I'm assuming in the second diagram the shape would be spherical with the sensor at the center.