All Answers 358187 Answers

Maybe you should try using my method.

I have never had a problem.

OK, young person, here is my interpretation of your question:

3^(-1) mod 25 is the "modular multiplivative inverse of 3 mod 25".

3^(-2) =1 / 9, which can be written as 9^(-1) mod 25. So then, you have:

[3^(-1) + 9^(-1)] = 12^(-1) mod 25 = 23 - which is the modular multiplicative inverse of 12 mod 25.

Thanks for the help, CPhill!!!

Thanks for your great solution, Rom!!!

This might help you. Here are the first 10 rows of Pascal's Triangle and some rules about the rows of the Triangle:
1   = R 0
1   1   = R 1
1   2   1   = R 2
1   3   3   1   = R 3
1   4   6   4   1   = R 4
1   5   10   10   5   1   = R 5
1   6   15   20   15   6   1   = R 6
1   7   21   35   35   21   7   1   = R 7
1   8   28   56   70   56   28   8   1   = R 8
1   9   36   84   126   126   84   36   9   1   = R 9
1   10   45   120   210   252   210   120   45   10   1   = R 10
The sum of the nth row is 2^n. 
The sum of rows 0 through n is 2^(n + 1) - 1.
The generating function of the nth row is (x + 1)^n. 

and actually part D too, sorry!

by the way my equation is $C(n,0)^2$,$C(n,1)^2$,$C(n,2)^2$...$C(n,n)^2$=$C(2n,n)$

Do you have any help?

um nevermind..........

\( \mathbf{d} = \begin{pmatrix}d_1 \\ d_2 \\d_3 \end{pmatrix}\)

\(for \ the \ line \ through \ B = (1, 1, 2)\ and \ C= (2, 3, 1) \ such \ that \ d_1 + d_2 + d_3 = 10.\)

A direction vector  =    (2 - 1, 3 - 1, 1 - 2)  =     ( 1, 2 , -1)

So....the sum of the components  of the direction vector  =  2

This implies that if we multiply each component by 5, then the sum of the components  = 10

So  we have  5( 1, 2 , - 1)  =   (5, 10, -5)      =   ( d1, d2, d3)

And the sum of  d1  + d2 + d3  =  10

The equation of the line can be

→

 r   =  (1,1,2)  + t (5, 10, - 5)

If t  = (1/5)   then  we get  ( 2, 3, 1)  =  C

Thanks, Alan.....that makes sense.....!!!!

→

AB  =   (B) - (A)  =  (2, 3 , 1)  -   ( 1, 0 , -1)    =   vector ( 1, 3, 2)

→

AC  =   (C) - (A)  =  ( 0, 1 , 1 ) - (1, 0 , - 1)  =   vector  ( -1, 1, 2)

→

BC  =  (C) - (B)   =  ( 0, 1 , 1)  -  (2, 3, 1)  =  vector   ( -2, -2, 0 )

See the following image:

Let vectors u, v and w  be three edges of the paralleliped  which meet at  (0,0, 0)

So.....these vectors  can be :

  u  = (3, 0 , 0)     v  =  (2,0,5)     w  = (0,5, 1)

The volume of the  parallelpiped   =   the absolute value  of   the  determinant  of the following matrix

u1   u2   u3                3  0   0                  [ 0 5 ]              [ 2  5 ]           [ 2   0 ]

v1   v2    v3   =          2  0   5      =   3  *  [ 5  1 ]   +  0 * [ 0  1 ]   +  0 *[ 0   5 ]   =

w1  w2   w3               0   5  1

3 [ 0*1  - 5 *5 ] + 0 + 0   =   3 [ -25 ]   =   - 75   =  l -75 l   =  75  units^3

Note that x-y must be larger than x+y if the ratio is to be between 2 and 5. This will be the case if y is negative.

So we have (x-y)/(x+y) > 2,  or. x-y > 2x+2y or x < -3y or x/y > -3 where the < becomes > because we are dividing by a negative number.

We also have (x-y)/(x+y)<5,  or x-y < 5x+5y or 4x > -6y or x > -(3/2)y or x/y < -3/2 where > becomes < because we are dividing by a negative number.

Hence -3 < x/y < -3/2, giving x/y = -2

Max got the right answer with incorrect reasoning! It happens!

if you don't get it the dude is a slave from the past and i skips over to the future

a^2 + b^2  = 325^4

 WolframAlpha shows 

https://www.wolframalpha.com/input/?i=a%5E2+%2B+b%5E2%C2%A0+%3D+325%5E4

this is the picture Omi 67 putt.

One question, Max

If  the solution is  -3/2 < z < -3

What integer  is > -3/2   and < -3  ???.....it can't be  z =   -2

y  = ( x^3 + 8x^2 + 21x + 18) / (x + 2)

Using polynomial division to find the other polynomial factor, we have

              x^2  + 6x  + 9

x + 2   [  x^3 + 8x^2  + 21x  + 18 ]

              x^3 + 2x^2  

              ____________________

                         6x^2 + 21x

                         6x^2 + 12x

                         ________________

                                       9x  + 18

                                       9x  + 18

                                      ________

Then  we have that   y  = ( x^3 + 8x^2 + 21x + 18) / (x + 2)  =   (x^2 + 6x + 9) (x + 2) / (x + 2) =

x^2 + 6x + 9

The discontinuity occurs at   x + 2  = 0 ⇒   x  =  -2  = D

And A  = 1   B  = 6    C  = 9

So   A  + B + C + D    =   14

Here is a graph of this :https://www.desmos.com/calculator/nnhk1oofs1

A right circular cylinder with radius 2 is inscribed in a hemisphere with radius 5 so that its bases are parallel to the base of the hemisphere. What is the height of this cylinder?

https://free-picload.com/images/2019/07/30/214.jpg

We can use the Pythagorean Theorem to find the height of the cylinder 

We have

height  = √ [ 5^2 - 2^2 ]  = √[25 - 4]   =  √21  units

Here is a cross-section of this : 

\(\text{Let }\dfrac{x}{y}=z.\\ 2 < \dfrac{x - y}{x + y} < 5\\ 2 < \dfrac{z - 1}{z + 1} < 5\\ 2z + 2 \stackrel{\boxed{1}}{<} z - 1 \stackrel{\boxed{2}}{<} 5z + 5\\ \boxed{1}: 2z + 2 < z - 1\\ z < -3\\ \boxed{2}: z - 1 < 5z + 5\\ -6 < 4z\\ z > -\dfrac{3}{2}\\ \text{Combine }\boxed{1} \text{ and }\boxed{2}: -\dfrac{3}{2} < z < -3\\ \text{The only integer solution is } z = -2\\ \therefore \dfrac{x}{y} = -2\)

See this link:  https://www.wolframalpha.com/input/?i=Give+integer+solutions+++++%5B++2+%3C+%5Cfrac%7Bx+-+y%7D%7Bx+%2B+y%7D+%3C+5+%5D