**Let a and b be coprime integers, and let m be an integer such that a | m and b | m. Prove that ab | m**

2022-04-04

**Let a and b be coprime integers, and let m be an integer such that a | m and b | m. Prove that ab | m**

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asked 2022-03-27

Deriving the confidence interval

$P(-{\mathrm{\Phi}}_{\alpha}^{-1}<X<{\mathrm{\Phi}}_{\alpha}^{-1})=1-2\alpha$

Let $P\sim \mathcal{N}(0,1)$

$\begin{array}{rlrlr}& P(-{\mathrm{\Phi}}_{\alpha}^{-1}X{\mathrm{\Phi}}_{\alpha}^{-1})& & & \left(1\right)\\ & P(X{\mathrm{\Phi}}_{\alpha}^{-1})-P(X-{\mathrm{\Phi}}_{\alpha}^{-1})& & P(AXB)=P(XB)-P(XA)& \left(2\right)\\ & \mathrm{\Phi}\left({\mathrm{\Phi}}_{\alpha}^{-1}\right))-\mathrm{\Phi}(-{\mathrm{\Phi}}_{\alpha}^{-1}\left)\right)& & P(XA)=\mathrm{\Phi}\left(A\right)& \left(3\right)\\ & \mathrm{\Phi}\left({\mathrm{\Phi}}_{\alpha}^{-1}\right))-(1-\mathrm{\Phi}\left({\mathrm{\Phi}}_{\alpha}^{-1}\right)\left)\right)& & \mathrm{\Phi}(-x)=1-\mathrm{\Phi}\left(x\right)& \left(4\right)\\ & \alpha -1+\alpha & & f\left({f}^{-1}\right)=id& \left(5\right)\\ & {-}{1}\times (1-2\alpha )& & & \left(6\right)\end{array}$

asked 2022-03-29

A nonhomogeneous second-order linear equation and a complementary function ${y}_{c}$ are given. Find a particular solution of the equation.

$4{x}^{2}y{}^{\u2033}-4x{y}^{\prime}+3y=8{x}^{\frac{3}{4}}$

asked 2022-03-29

Find a vector function that represents the curve of intersection of the two surfaces of the cylinder x^2+y^2=4 and the surface z=xy.

asked 2022-04-30

A quadratic function has its vertex at the point (−7,2)$(-7,2)$. The function passes through the point (8,3)$(8,3)$. When written in vertex form, the function is f(x)=a(x−h)2+k$f\left(x\right)=a{(x-h)}^{2}+k$, where:

asked 2022-04-28

Sally has caught covid but doesn’t know it yet. She is testing herself with rapid antigen kits which have an 80% probability of returning a positive result for an infected person. For the purpose of this question you can assume that the results of repeated tests are independent.

a) If sally uses 3 test kits what is the probability that at least one will return a positive result?

b) In 3 tests, what is the expected number of positive results?

c) Sally has gotten her hands on more effective tests, these ones have a 90% probability of returning a positive result for an infected person. If she tested herself

twice with the new tests, how many positive results would she expect to see?

asked 2022-05-10

Use the linear approximation of Use the linear approximation of f(x,y)=e2x2+3y at (0,0) to estimate f(0.01,−0.02). at $(0,0)$ to estimate $f(0.01,-0.02)$.

asked 2022-03-24

Given an sample of data set X which is normal-distributed to $N(\mu =240,\sigma )$, I want to find the $95\mathrm{\%}$ confidence interval of $max(\mu -200,0)$. As the $max(\mu -200,0)$ cut the normal-distribution curve at the point of 200, the sample of $max(\mu -200,0)$ is not more normal distributed. Therefore what is a reasonable $95\mathrm{\%}$ confidence interval?