Find the smallest positive integer k satisfying the stated condition or explain no such k exists:

${3}^{k}\equiv 1\text{mod}11$ .

alesterp
2021-01-28
Answered

Find the smallest positive integer k satisfying the stated condition or explain no such k exists:

${3}^{k}\equiv 1\text{mod}11$ .

You can still ask an expert for help

unett

Answered 2021-01-29
Author has **119** answers

Step 1

Since gcd(3,11)=1 and again by using Eulers

Since gcd(3,11)=1 and again by using Eulers

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I need help with the following optimization problem

$max\phantom{\rule{thickmathspace}{0ex}}\alpha \mathrm{ln}(x(1-{y}^{2}))+(1-\alpha )\mathrm{ln}(z)$

where the maximization is with respect to $x,y,z$, subject to

$\begin{array}{rl}\alpha x+(1-\alpha )z& ={C}_{1}\\ \alpha y\sqrt{x(x+\gamma )}-\alpha x& ={C}_{2}\end{array}$

where $0\le \alpha \le 1$, $\gamma >0$, and $x,z\ge 0$, and $|y|\le 1$.

Generally, one can substitute the constraints in the objective function and maximize with respect to one parameter. The problem is that in this way things become algebraically complicated, and I believe that there is a simple solution.

$max\phantom{\rule{thickmathspace}{0ex}}\alpha \mathrm{ln}(x(1-{y}^{2}))+(1-\alpha )\mathrm{ln}(z)$

where the maximization is with respect to $x,y,z$, subject to

$\begin{array}{rl}\alpha x+(1-\alpha )z& ={C}_{1}\\ \alpha y\sqrt{x(x+\gamma )}-\alpha x& ={C}_{2}\end{array}$

where $0\le \alpha \le 1$, $\gamma >0$, and $x,z\ge 0$, and $|y|\le 1$.

Generally, one can substitute the constraints in the objective function and maximize with respect to one parameter. The problem is that in this way things become algebraically complicated, and I believe that there is a simple solution.

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These two triangles are similar by AA similarity. The ratio of corresponding altitudes is proportional to the ratio of corresponding sides. Find the value of x.

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To prove: The two triangles are similar using postulate or theorem.

Given information:

The system triangles:

Given information:

The system triangles:

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Show that the two triangles are similar.

$\mathrm{\angle}VRS\stackrel{\sim}{=}\mathrm{\angle}\text{}?\text{}{\textstyle \phantom{\rule{1em}{0ex}}}\text{and}{\textstyle \phantom{\rule{1em}{0ex}}}\text{}\mathrm{\angle}SVR\stackrel{\sim}{=}\mathrm{\angle}\text{}?,\text{}so\text{}\mathrm{\u25b3}SVR\sim \mathrm{\u25b3}TRU.$

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To prove:The two triangles are similar using postulate or theorem.

Given information:

The system triangles:

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Describe one similarity and one difference between the definitions of $\mathrm{sin}0{\textstyle \phantom{\rule{1em}{0ex}}}\text{and}{\textstyle \phantom{\rule{1em}{0ex}}}\mathrm{cos}0$ , where 0 is an acute angle of a right triangle.

asked 2022-08-06

What is the probability that the determinant of a matrix of order 2 is positive ,whose elements are i.u. ran.var.in the interval(0,1)?

I have been thinking about the problem for quite a while and it seems to me a problem on geometric probability albeit in four dimensions.For if the elements of the matrix are a,b,c,d(row-wise) the value of the determinant is ad-bc.the problem reduces to finding the probability P(ad-bc>0) subject to given domain and distributation of a,b,c,d.In our problem the sample space is the 4D unit cube.If we take the axes as X,Y,Z,U THE REQUISITE PROBABILITY IS GIVEN BY THE FRACTION OF 4D VOLUME OF THE UNIT CUBE FOR WHICH XY-ZU>0. I would love to see a solution along these goemetrical lines.

I have been thinking about the problem for quite a while and it seems to me a problem on geometric probability albeit in four dimensions.For if the elements of the matrix are a,b,c,d(row-wise) the value of the determinant is ad-bc.the problem reduces to finding the probability P(ad-bc>0) subject to given domain and distributation of a,b,c,d.In our problem the sample space is the 4D unit cube.If we take the axes as X,Y,Z,U THE REQUISITE PROBABILITY IS GIVEN BY THE FRACTION OF 4D VOLUME OF THE UNIT CUBE FOR WHICH XY-ZU>0. I would love to see a solution along these goemetrical lines.