Recent Questions - Mathematics Stack Exchange |
- Counterexamples for Banach *-algebras
- Inclusive vs Exclusive OR: Tomorrow it ill rain or it will be dry all day?
- probability question on drawing ball
- Derivative of a state variable as an output of an affine system
- Question about a remark in Serre's Linear Representations of Finite Groups
- Fractal dimension of domain wall in Ising model at criticality
- Different proofs for $A ({\rm adj}A)=\det(A)I$
- Finding $\int^1_0 \frac{x\,\mathrm dx}{((2x-1)\sqrt{x^2+x+1} + (2x+1)\sqrt{x^2-x+1})\sqrt{x^4+x^2+1}}$
- If $E|_k$ is a finite extension such that $E^pk=E$, then prove $E^{p^m}k=E$ where $p=$char$(k)$
- Find the mistake in my conditional probability calculation
- Creating a D- dimensional array and evaluating a function
- Discrete Turning Number
- Find the loan amount borrowed
- Assume Elliott-Halberstam conjecture, then prove a ANT inequality.
- How to write this sum $\sum_{m=0}^\infty (2m+1)^{-2k-1} \sum_{r=1}^\infty (-1)^{r-1} e^{-2(2m+1)r a} $ as a sum over single index?
- Normalizing the columns of a matrix through matrix multiplications
- What is the probability that at least two spinner land on Red
- How would you reduce the Tsiolkovsky Rocket Equation to one paragraph?
- Is there a quartic or quintic formula?
- Integral $\int^{x=\frac{\pi}{4}}_{x=0} \int^{y= \cos{x}}_{y=\sin{x}} dydx$
- Is the inequality $\sqrt{(2^r + 1)(t + 1)} \leq \min(2^r, t)$ false in general?
- What is the limit of $\frac{\sin 2x -2\sin x}{x^3}$ when $x$ tends to $0$?
- Definition of $\mathcal O_X$-derivation on scheme $X$ in Vakil
- Clebsch-Gordan coefficients for the real spherical harmonics
- How to know if a system of equations of the form $ y_i = \sum_{j=0}^{n} c_j e^{jx_i}$ is solvable
- How to find the total loan amount borrowed? [closed]
- About the inequality $x^{x^{x^{x^{x^x}}}} \ge \frac12 x^2 + \frac12$
- Geometric progression question. Year less?
- Closed form for CRT solution using Euler $\phi\,$ (totient)
- If $\gcd(a,b) = 1$ and $a,b\mid x$ then $ab\mid x$.
| Counterexamples for Banach *-algebras Posted: 12 Jun 2021 10:55 PM PDT
These are not homework problems. Such examples don't exists for a $C^*$-algebra and I am unable to prove these in a Banach $*$-algebra.  |
| Inclusive vs Exclusive OR: Tomorrow it ill rain or it will be dry all day? Posted: 12 Jun 2021 10:51 PM PDT I'm taking Keith Devlin's course "Introduction to Mathematical Thinking". Currently, I am learning Exclusive-OR and Inclusive-OR. Keith provides one factoid, that, Exclusive-OR statements can be formatted using "Either-OR". Moving further he provides an example.
Which he indicates that above statement is Inclusive-OR. But then, he provides following statement and asks student whether it is Inclusive or Exclusive OR.
Is it same as saying,
And answer to this questions is "Exclusive-OR. My question is, given a sentence using 'or', how to determine that it is Inclusive or Exclusive OR? Because, I think that given limited information it is difficult to determine inclusive or exclusive form of OR.  |
| probability question on drawing ball Posted: 12 Jun 2021 10:57 PM PDT
My solution: as the last ball is red, that means we have drawn $19$ red balls and $16$ blue balls so far. The total no of ways of doing it is $35 \choose19$. Now total no of ways of drawing all the balls is $36 \choose 20$. The probability is $$\frac{35\choose19}{36\choose20}=\frac{5}{9}$$ But the answer is $\frac{4}{9}$. Can you please correct my solution?  |
| Derivative of a state variable as an output of an affine system Posted: 12 Jun 2021 10:41 PM PDT Given simple system of ODE. \begin{cases} \dot{x_1}=-x_1+u \\ \dot{x_2}=-x_2-x_1 \end{cases} As an output, I want to use $y=\dot{x_1}$. I'm trying to convert to an affine state-space, but in the case of such an output, descriptor matrix is singular (code in Mathematica for example). I got the idea to add a differentiating filter: \begin{cases} \dot{x_1}=-x_1+u \\ \dot{x_2}=-x_2-x_1 \\ \frac{1}{k}\dot{X}+X=\dot{x_1} \end{cases} And now $y=X \approx \dot{x_1}$ and $k>>1$. Are there some simpler ways to modify the equation and get $y=\dot{x_1}$?  |
| Question about a remark in Serre's Linear Representations of Finite Groups Posted: 12 Jun 2021 10:39 PM PDT I'm reading the section 11.1 in Serre's Linear Representations of Finite Groups. My question is about the existence of $\varphi$ in the remark. If $\varphi$ is the function mentioned in the remark, then $\varphi=\sum\limits_{H\in X}\varphi Ind_H^G f_H=\sum\limits_{H\in X}Ind_H^G (f_H Res_H^G\varphi)=\sum\limits_{H\in X}Ind_H^G (f_H \varphi_H)$. Thus we can construct $\varphi$ as $\sum\limits_{H\in X}Ind_H^G (f_H \varphi_H)$. But how to see that the constructed $\varphi$ satisfy that $Res_H^G \varphi=\varphi_H$? I tried direct computation, but I got nowhere. Since Serre didn't give a proof, I guess the result should be obvious. Maybe I'm missing something. Any help? Thanks!  |
| Fractal dimension of domain wall in Ising model at criticality Posted: 12 Jun 2021 10:38 PM PDT Consider the Ising model on an $L \times L$ lattice with periodic boundary conditions in the east/west directions and with spins on the north boundary fixed as $+1$ and the spins on the south boundary fixed as $-1$. Then, if you fix a cold temperature $T < T_c$, as $L \to \infty$ you expect that the domain wall between the $+1$ spin cluster on the top and the $-1$ spin cluster on the bottom approaches a flat line (dimension 1), and exactly at $T = T_c$ you expect there to be some nontrivial behaviour, where you end up with the domain wall being some fractal dimension between $1$ and $2$. This is just intuition I've gleaned from a talk I attended ~2 years ago, but I can't find any references following this line of thought. Are there any places to look to see experiments in this direction? In particular, I'm mainly curious how to rigorously define what the fractal dimension we are measuring at $T = T_c$ is in the thermodynamic limit, and what this number turns out to be. As an example of how to formalise what I'm talking about, we can consider the box-counting dimension, and simply define the fractal dimension of the domain wall to be: $$\text{dim}_\text{box}(\text{Domain Wall}) = \lim_{\epsilon = \frac{1}{L} \to 0} \frac{\langle\text{Number of sites that the domain wall passes through}\rangle_\text{Gibbs Distribution on L x L lattice}}{\log(1/\epsilon)}$$ There might be other formulations based on the actual thermodynamic limit measure, but this isn't something I'm too familiar with  |
| Different proofs for $A ({\rm adj}A)=\det(A)I$ Posted: 12 Jun 2021 10:38 PM PDT If $A$ is an $n\times n$ matrix, then ${\rm adj}(A)$ is the the $n\times n$ matrix, whose $(j,i)$-th entry is $(-1)^{i+j}$ times the determinant of a submatrix of $A$ obtained by deleting $i$-th row and $j$-th column. After introduction of ${\rm adj}(A)$, one immediately learns the most important result that $$A{\rm adj}(A)=\det(A)I.$$ Now for proof of this, one way is to proceed with the definition of $\det(A)$ by expansion along any row or column. Are there any other proofs of this result? I was unable to find in any books.  |
| Posted: 12 Jun 2021 10:49 PM PDT $$\int^1_0 \frac{x\,\mathrm dx}{((2x-1)\sqrt{x^2+x+1} + (2x+1)\sqrt{x^2-x+1})\sqrt{x^4+x^2+1}}$$ I've tried :
As per wolframAlpha, a beautiful elementary closed form exists $$ \frac{\sqrt{\left(x^4+x^2+1\right)}}{3}\left(\frac{1}{\sqrt{\left(x^2-x+1\right)}}- \frac{1}{\sqrt{\left(x^2+x+1\right)}}\right) $$ I'll be happy with solution of just definite integral, bonus points if you get closed form solution too.  |
| If $E|_k$ is a finite extension such that $E^pk=E$, then prove $E^{p^m}k=E$ where $p=$char$(k)$ Posted: 12 Jun 2021 10:36 PM PDT Here $E^p:=\{x^p:\ x\in E\}$. As $k$ is field of charateristic $p$, $E^p$ is a subfield of $E$. Now given that $E^pk=E$. I'm trying to show that $E^{p^m}k=E$. If I can prove that $E^pE^p=E^{2p}$. Then It will follow that $E^{2p}k=E^pE^pk=E^pE=E$. By induction $E^{np}k=E$, hence $E^{p^m}k=E$. But I'm unable to prove $E^pE^p=E^{2p}$. Then how to argue $E^{p^m}k=E$. This might look easy but I'm not getting it. Can anyone help me in this regard? Thanks for help in advance.  |
| Find the mistake in my conditional probability calculation Posted: 12 Jun 2021 10:52 PM PDT A meeting has $12$ employees. Given that $8$ of the employees are women, find the probability that all the employees are women? I just defined the two events $A : 8$ employees are female and $B :$ all employees are female. Thus, we need $P(B|A)=\frac{P(A \cap B)}{P(A)}$. Now, we know that $P(A \cap B) = P(B)$. So, $P(B)=\frac{1}{2^{12}}$ considering equal probability of male and females. Also, $P(A)={12 \choose 8} \frac{1}{2^{12}}$. On dividing the conditional probability comes out to be $\frac{1}{{12 \choose 8}}$ but it does not match with the answer to this question. Where am I going wrong? Anyone please help!  |
| Creating a D- dimensional array and evaluating a function Posted: 12 Jun 2021 10:34 PM PDT I am new to Matlab and could use some help. I want to create a d-dimensional box and evaluate a function $f(x) = \sum_i^d x_i$ over the grid. I am encountering some issues while coding I did some reading and experimentation using $ndgrid$ from Matlab, but I think I am not using it correctly as I get a list and then I cannot evaluate it. Basically I created $x = linespace(0,1,N)$ then I wanted to create a function that takes the dimension d and x and give me the mesh grid that I can pass to f, and get the function evaluations. Thank you very much for your time guys.  |
| Posted: 12 Jun 2021 10:34 PM PDT There are two equivalent way to define turning number of a immersed closed curve.
For example, according to wikipedia, we have turning number $3$ for the curve below:
Meanwhile, the turning number of a polygonal closed path is defined to be the total discrete curvature of the path dividing $2\pi$. Namely, $\dfrac{\sum_{i=1}^n(\pi-\alpha_i)}{2\pi}$, where $\alpha_i$ are the "internal angles" of the polygon. Note we define interior angles to be the angles you get when you FOLLOW the polygon (the angle could be on the exterior of the shape). Exterior angles are simply the complement of the interior ones. See Theorem 2.1, p.31. I think the turning number of a polygonal closed path can also defined to be the number of full rotation of $2\pi$ as walking along the polygon and turning at the vertices.
For example, there are two anticlockwise full rotation of $2\pi$ and one clockwise full rotation of $2\pi$ in the left of the above figure. This gives $2(+1)+(-1)=1$ as the turning number. While there is one anticlockwise full rotation of $2\pi$ in the right of the above figure. This gives $+1$ as the turning number.
 |
| Posted: 12 Jun 2021 10:43 PM PDT I am working on the EMI calculator module as part of the current banking project. I find answers for calculating simple interest, EMI, and even tenure but not able to find a formula to calculate the total amount borrowed. I found the below formula to calculate what I want but that is incorrect source: Find Loan Amount using parameters ROI and Tenure Here are the loan details, Monthly EMI: 32530, Loan tenure: 5 years, Interest: 10.90 Total loan amount: ? What is the formula to find the total loan amount? What i have tried,
Here is the formula i have taken from the above source,
Here is the source  |
| Assume Elliott-Halberstam conjecture, then prove a ANT inequality. Posted: 12 Jun 2021 10:36 PM PDT Here is Elliott-Halberstam conjecture. $$ \sum_{q\leqslant x^{1-\varepsilon}}{\underset{y\leqslant x}{\max}\underset{\begin{array}{c} 1\leqslant a\leqslant q\\ \left( a,q \right) =1\\ \end{array}}{\max}\left| \psi \left( y,q,a \right) -\frac{y}{\varphi \left( q \right)} \right|\ll x}\log ^{-A}x $$ Then show $$ \sum_{2\leqslant p\leqslant x}{\tau _3\left( p-1 \right)}\ll x\log x $$ is true. Here $$ \tau _3\left( x \right) =\sum_{d\left| x \right.}{d^3} $$ I tried several methods but I don't know how to apply the conjecture appropriately. Looking for hint or answer in detail.  |
| Posted: 12 Jun 2021 10:48 PM PDT So I want to write the sum $$\sum_{m=0}^\infty (2m+1)^{-2k-1} \sum_{r=1}^\infty (-1)^{r-1} e^{-2(2m+1)r a} $$ where $a>0$ and $k\in \mathbb{N}$, as a sum over single index which probably uses odd divisors of $(2m+1)\cdot r$. However, I am quite confused about this method in general. I am trying to learn this concept. If anyone can help me with the logic, it will be highly appreciated.  |
| Normalizing the columns of a matrix through matrix multiplications Posted: 12 Jun 2021 10:26 PM PDT Generally the normal equation is derived using a calculus-based approach of minimizing the least squares error. I'm trying to learn the linear algebra approach. My understanding so far is as follows: Consider the standard linear regression problem: $Ab=y$ or $\sum_iA_{(i)}b^i=y$, where $A$ is an $m\times n$ matrix with $m>n$, and $A_{(i)}$ is the $i$-th column of $A$. Assuming $A$ to be full rank, its columns form a basis $C$ of an $n$-dimensional subspace of $\mathbb{R}^m$. We already know the standard $\mathbb{R}^m$ basis representation of $y$ and we are trying to find its representation in the $C$ basis, such exact representation is possible to find only if $y\in\text{span}(C)$. Most likely $y$ won't lie in $\text{span}(C)$. In that case, the most natural alternative would be to find the $C$-representation of the next best thing - the projection of $y$ in $\text{span}(C)$. In other words, I need to find $$\hat b^i=\frac{\langle y,A_{(i)}\rangle}{\|A_{(i)}\|}$$ Now if only I had a matrix $\tilde A$ whose columns are the normalized versions of the columns of $A$, i.e., $\tilde A_{(i)}=A_{(i)}/\|A_{(i)}\|$, then I can just write $\hat b=\tilde A^Ty$. The question is: is it possible to get $\tilde A$ from multiplication/transpose/inverse operations performed on $A$?  |
| What is the probability that at least two spinner land on Red Posted: 12 Jun 2021 10:37 PM PDT Three spinners are marked with equal amounts of Red, Blue and Yellow. At a particular instance, all three are spun together. What is the probability that at least two of the spinners land on red? The at least part is confusing me. My attempt: So if all three lands on red the probability will be: $P(All\space red)=\frac{1}{3}\times\frac{1}{3}\times\frac{1}{3}=\frac{1}{27}$ Having at least two will have a higher probability; two of the spinners should have red and the other can have any colour: $$P(Two\space red\space and\space one\space any)=\frac{1}{3}\times\frac{1}{3}\times\frac{3}{3}=\frac{1}{9}$$  |
| How would you reduce the Tsiolkovsky Rocket Equation to one paragraph? Posted: 12 Jun 2021 10:24 PM PDT I'm working on a project where I must describe the Tsiolkovsky Rocket Equation in a short amount of time. As such, I will have to use only the vital parts of the equation and its usages. Now, I have been researching non-stop (!) about the equation for the past week or so, and have a "sort of" grasp on what it entails. However, I am curious about what YOU all think of it. There could easily be vital points which I miss out on, which I would not want to do. How would you reduce the rocket equation, or describe it, in just one sentence or paragraph? (And don't worry, this isn't a text-based project. I won't plagiarize your explanations! ^^;) I'm not sure if I should be asking this here or on the Physics stackexchange. But still, thank you all in advance! Very curious to see a paraphrased explanation of this rather large subject.  |
| Is there a quartic or quintic formula? Posted: 12 Jun 2021 10:53 PM PDT I know about the quadratic formula, and the cubic formula, so I was wondering if there were any more. My teacher said there was no such thing as a quintic formula, so I was wondering that if there was no such thing as one, why. Thank you.  |
| Integral $\int^{x=\frac{\pi}{4}}_{x=0} \int^{y= \cos{x}}_{y=\sin{x}} dydx$ Posted: 12 Jun 2021 10:55 PM PDT $$\int^{x=\frac{\pi}{4}}_{x=0} \int^{y= \cos{x}}_{y=\sin{x}} dydx$$ I got the answer $\sqrt{2} - 1 $ but my tutor got $8$? I assumed that I am starting with integrating $1 dydx$, that is how I got my answer. Is my answer wrong?  |
| Is the inequality $\sqrt{(2^r + 1)(t + 1)} \leq \min(2^r, t)$ false in general? Posted: 12 Jun 2021 10:55 PM PDT My question here is pretty basic:
I tried asking WolframAlpha for its solutions, it says that no solutions exist. I also tried asking WolframAlpha for the solutions to the closely related inequality $$\sqrt{(2^r + 1)(t + 1)} \leq \max(2^r, t),$$ it gave the following solutions: $$t \geq \frac{\sqrt{2^{2r} + 3 \times {2^{r+1}} + 5} + 2^r + 1}{2}$$ and $$-1 \leq t \leq \frac{-2^r + 2^{2r} - 1}{2^r + 1}.$$ Note that the first solution is for $t > 2^r$ (and therefore, $\max(2^r, t) = t$), and that the second solution is for $t < 2^r$ (and therefore, $\max(2^r, t) = 2^r$). Finally, here is the inequality plot for $$\sqrt{(2^r + 1)(t + 1)} \leq \max(2^r, t):$$
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| What is the limit of $\frac{\sin 2x -2\sin x}{x^3}$ when $x$ tends to $0$? Posted: 12 Jun 2021 10:51 PM PDT $$\lim_{x\to 0} \frac{\sin2x -2\sin x}{x^3}$$ I am getting two values for this limit. Can anybody please share their solution so I can cross check my answers? I see all your answers and yes, I have tried this method. Here is the other method I was talking about $\lim_{x\to 0} \frac{\sin2x -2\sin x}{x^3}$ Using the formula $\lim_{x\to 0} \frac{\sin x}{x} = 1$ $\lim_{x\to 0} \frac{2}{x^2} - \frac{2}{x^2}$ Which gives the value 0  |
| Definition of $\mathcal O_X$-derivation on scheme $X$ in Vakil Posted: 12 Jun 2021 10:29 PM PDT I'm reading Vakil's note about differential sheaf. Let $\pi:X\rightarrow Y$ be a morphism of schemes. He defines $d:\mathcal O_{X}\rightarrow \Omega_{X/Y}$ as follow. Let $pr_1:X\times_YX\rightarrow X$ and $pr_2:X\times_YX\rightarrow X$ be projections. Then $d$ on open set $U$ is defined by $df = pr_2^*f - pr_1^*f$. My question:
Thank you in advance!  |
| Clebsch-Gordan coefficients for the real spherical harmonics Posted: 12 Jun 2021 10:47 PM PDT There are many online calculators for the Clebsch-Gordan coefficients of the complex spherical harmonics, and many text books on quantum mechanics include tables of Clebsch-Gordan coefficients. I wonder about Clebsch-Gordan coefficients for the real spherical harmonics. Can they be related in some way to the coefficients for the complex spherical harmonics? If not, how can they be derived?  |
| How to know if a system of equations of the form $ y_i = \sum_{j=0}^{n} c_j e^{jx_i}$ is solvable Posted: 12 Jun 2021 10:55 PM PDT I was working on a problem and faced a this system of equations ($y_i$ and $x_i$ are givens) $$ y_i = \sum_{j=0}^{n} c_j e^{jx_i} \quad0 \le i \le n$$ is there a way to determine this system is solvable or not?  |
| How to find the total loan amount borrowed? [closed] Posted: 12 Jun 2021 10:26 PM PDT I am working on the EMI calculator module as part of the current banking project. I find answers for calculating simple interest, EMI, and even tenure but not able to find a formula to calculate the total amount borrowed. I found the below formula to calculate what I want but that is incorrect source: Find Loan Amount using parameters ROI and Tenure Here are the loan details, Monthly EMI: 32530, Loan tenure: 5 years, Interest: 10.90 Total loan amount: ? What is the formula to find the total loan amount? What i have tried,
based on the formula, where 𝐴=EMI (monthly payment)𝑖=interest𝑚=type of installment (12 = monthly, 1 = annually)𝑡=tenure (time)  |
| About the inequality $x^{x^{x^{x^{x^x}}}} \ge \frac12 x^2 + \frac12$ Posted: 12 Jun 2021 10:35 PM PDT
Remark 1: The problem was posted on MSE (now closed). Remark 2: I have a proof (see below). My proof is not nice. For example, we need to prove that $\frac{3x^2 - 3}{x^2 + 4x + 1} + \frac{12}{7} - \frac{24x^{17/12}}{7x^2 + 7} \le 0$ for all $0 < x < 1$ for which my proof is not nice. I want to know if there are some nice proofs. Also, I want my proof reviewed for its correctness. Any comments and solutions are welcome and appreciated. My proof (sketch): We split into cases: i) $x \ge 1$: Clearly, $x^{x^{x^{x^{x^x}}}}\ge x^x$. By Bernoulli's inequality, we have $x^x = (1 + (x - 1))^x \ge 1 + (x - 1)x = x^2 - x + 1 \ge \frac12 x^2 + \frac12$. The inequality is true. ii) $0 < x < 1$: It suffices to prove that $$x^{x^{x^{x^x}}}\ln x \ge \ln \frac{x^2 + 1}{2}$$ or $$x^{x^{x^{x^x}}} \le \frac{\ln \frac{x^2 + 1}{2}}{\ln x}$$ or $$x^{x^{x^x}}\ln x \le \ln \frac{\ln \frac{x^2 + 1}{2}}{\ln x}$$ or $$x^{x^{x^x}}\ge \frac{1}{\ln x}\ln \frac{\ln \frac{x^2 + 1}{2}}{\ln x}.$$ It suffices to prove that $$x^{x^{x^x}}\ge \frac{7}{12} \ge \frac{1}{\ln x}\ln \frac{\ln \frac{x^2 + 1}{2}}{\ln x}. \tag{1}$$ First, it is easy to prove that $$x^x \ge \mathrm{e}^{-1/\mathrm{e}} \ge \frac{1}{\ln x}\ln\frac{\ln\frac{7}{12}}{\ln x}.$$ Thus, the left inequality in (1) is true. Second, let $f(x) = x^{7/12}\ln x - \ln \frac{x^2 + 1}{2}$. We have \begin{align*} f'(x) &= \frac{7}{12x^{5/12}} \left(\ln x + \frac{12}{7} - \frac{24x^{17/12}}{7x^2 + 7}\right)\\ &\le \frac{7}{12x^{5/12}} \left(\frac{3x^2 - 3}{x^2 + 4x + 1} + \frac{12}{7} - \frac{24x^{17/12}}{7x^2 + 7}\right)\\ &\le 0 \tag{2} \end{align*} where we have used $\ln x \le \frac{3x^2 - 3}{x^2 + 4x + 1}$ for all $x$ in $(0, 1]$. Also, $f(1) = 0$. Thus, $f(x) \ge 0$ for all $x$ in $(0, 1)$. Thus, the right inequality in (1) is true. We are done.  |
| Geometric progression question. Year less? Posted: 12 Jun 2021 10:13 PM PDT Here is the math question.
This is how I worked out the question. Which is However my teacher said that it is wrong (to the whole class and also insulted me a bit >_>) and that I didn't follow this formula. And that I should have done Which is instead This doesn't make sense because if you used the formula and the question was
That doesn't make sense at all! Am I right or is the teacher right? ADDITIONALLY my teacher said his answer is an interpretation of the question. Is his answer a valid interpretation of this question? Or is it just incorrect mathematics? BTW Also my teacher said my answer has no common sense and that I won't be able to do the HSC well if I keep reading questions wrong.  |
| Closed form for CRT solution using Euler $\phi\,$ (totient) Posted: 12 Jun 2021 10:42 PM PDT Can anyone show some hints to this? If $\gcd(a,b)=1$, then $a^{\phi(b)}+b^{\phi(a)}=1 \pmod{ab}$. I know that $a^{\phi(b)}=1 \pmod{b}$, and similarly, $b^{\phi(a)}=1 \pmod{a}$, but then how do I combine the work so that I get the result I need? Thanks!  |
| If $\gcd(a,b) = 1$ and $a,b\mid x$ then $ab\mid x$. Posted: 12 Jun 2021 10:41 PM PDT If $\gcd(a,b) = 1$ and $a,b\mid x$ then $ab\mid x$. My attempt at answering the question: \begin{align*} x &\equiv 0 \pmod{a}\\\ &\Longrightarrow x\text{ is divisible by $a$}\\\ &\Longrightarrow x = ma\text{ for some integer $m$}\\\ \ \\\ x &\equiv 0 \pmod{b}\\\ &\Longrightarrow x\text{ is divisible by $b$}\\\ &\Longrightarrow x = mb\text{ for some integer $m$}\\\ \ \\\ x^2 &= (ma)(mb)\\\ x^2 &= (m^2)(ab)\\\ x &= \sqrt{m^2ab}\\\ x &= m\sqrt{a}\sqrt{b} \end{align*} Let $m$ be $k\sqrt{a}\sqrt{b}$. Then \begin{align*} x &= kab\\\ &\Longrightarrow x \equiv 0 \pmod{ab} \end{align*} Is this correct, if not can someone point me in the right direction?  |
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