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a = 1   and   b = 9

a + b = 1 + 9 = 10

If I understand your question:

"The sum of modulo 16 inverses of the first 4 positive odd integers" is:[1, 3, 5, 7]^ -1 mod 16 =[1 + 11 + 13 + 7] ==32

32 mod 16==0

sorry but it is incorrect

#1 is a load of nonsense.

\(\displaystyle 1-(a+bi)(c+di) \text{ is not the conjugate of }1+(a+bi)(c+di), \\ \text{and their product is not a real number.}\)

In addition, the expansion of the numerator is also not correct. Again the product is not a real number.

Here's a different method.

Take a line parallel to DE, a half a unit to the right.

It's the perependicular bisector of AC so will pass through the centre of the circle.

Drop a perpendicular from the centre of the circle onto DE, bisecting it.

Then, by Pythagoras, r^2 = 4^2 + (1/2)^2 = 16 + (1/4) = 65/4, so r = sqrt(65)/2.

Can you explain where th expresion for the radius comes from ?

Let  the center of the larger circle  be (10,10)....and this is the incenter of the large triangle 

Let A  be the angle at the  bottom right  vertex

The tangent of this angle is  4/3

But  the bisector of angle A   creates  an angle with 1/2 the measure of A

And the tangent of this angle is   sqrt   [  1 -cos A] / sqrt [1 + cos A]

And cos A = 3/5

So  tan (A/2)  =   sqrt  [ 1-3/5 ] / sqrt [1 + 3/5]   =  1/2

And

tan (A/2)   =10/20

cos (A/2)  =  2/sqrt (5)

Therefore, the distance  from the bottom right  vertex   of the large triangle to the  center of the larger circle can be found as

cos (A/2)  = 20 / D

D =  20 / (2 /sqrt (5) )  =  10sqrt (5) 

And using reflexive symmetry of similar polygons, the angle  bisector will go through the center of both  circles....

So  using similar triangles we  have that

10/ (10sqrt (5)  )  =  r / [ 10sqrt (5)  - 10  - r ]

1/sqrt (5)  =  r / [10sqrt (5)  -10  - r ] = 

r sqrt (5)  =  10sqrt (5)  - 10  - r   

r (  sqrt (5) + 1  )  =  10 ( sqrt (5) - 1)

r  =  10 ( sqrt (5) -1) ( sqrt (5) -1) / 4 

r = 10 ( 5 - 2sqrt (5) + 1) /4

r = 15 - 5sqrt (5)

Here's a pic  :

You're welcome!

We just need to solve this....let x  = 2  and we have that

k (2)^2  = k(2)  -20

4k  =  2k  -20

20 =  2k-4k

20 = -2k

k  = -10

See  the graph here  :   https://www.desmos.com/calculator/xfwuevr719

By the intersecting  chords theorem

EB * DB   =  AB * CB

EB * 2  =    3 * 4

EB    = 6

Angle EBC  =(1/2)  (minor arc EC + minor arc AD)

90  = (1/2)  ( minor arc  EC  + minor arc  AD)

180 =  minor arc EC + minor arc AD

EC^2  = 4^2 + 6^2   = 52

AD^2  =  3^2  + 2^2  =  13

Using the Law of  Cosines

EC^2 =  2 r^2 - 2r^2 cos A

AD^2  = 2r^2  - 2r^2  cos (180 - A)           where A is a central angle....simplify

Note.....cos (180 - A)  =  -cos A

So....the  second equation  becomes   

AD^2  = 2r^2 - 2r^2 (-cos A)    

13 = 2r^2  + 2r^2 cos A

So we have 

52  =2r^2  -2r^2 cos A

13  = 2r^2 +2r^2 cos A         add these equations

65 = 4r^2

65/4  = r^2  

r  = sqrt (65) / 2

2 * BE = 3 * 4   ==>  BE = 6            

Radius   r = sqrt(8² +1) / 2         r = √65 / 2

Total area = 400 * 500   = 200,000 ft^2

1/2 area  =   100,000 sq ft

Middle dimensions

(400-2x) ( 500 - 2x)   = 100000

200000 - 1800x  + 2x^2  = 100000

4x^2 - 1800x + 100000   =  0

2x^2   -900x  +  50000    = 0 

x ≈ 385 ft  (reject)  

or

x ≈  64.9 ft  = 65  ft

He needs to  make  about  65/ 5   = 13 passes

Let's see

1/5   [      5        -1        -30         6   ]

                          1            0       -6

          __________________________

               5         0        -30         0

Good job   !!!

Yes, but ONLY if her favorite number is of this form:

a(n) =2n^2. So, for numbers 1, 2, 3, 4, 5, 6, 7, 8,9, 10............etc

2*1^2 =2,    2*2^2=8,    2*3^2=18,    2*4^2 =32,    2*5^2=50............and so on.

This is much much bigger than your number:(9^9^9)! =((9^(387,420,489) ! =10^10^369,693,108.2

1/3, 1/8, 1/13 ......  

The formula for  the  nth term is

1 / [ 3 + 5(n -1) ] 

So

3 + 5(n -1)    =   123

5 ( n -1)   =120

n - 1  =   24

n =  25th term

Thank you bro!

1x^3  + 10x^2  + 27x  + 10

q                                     p

It's clear that the  root  must  be  negative

You have it a little confused

Remember  that p/q  =  all  the factors of 10/ all the  factors of  1

Possibilities

-1, -2. -5. -10

It can't be  -1

-2  [  1    10       27       10 ]

               -2      -16      -22

       __________________

       1        8      11       -12

-5  [ 1   10     27      10  ]

              -5    -25    -10

    __________________

       1       5    2          0

-5 is the integer  root

Is this from LMT? cuz it feels so familiar

Common ratio  =  -21/7 =  -3

8th term  =  7 (-3)^(n - 1)   =  7(-3)^(8-1)  = 7(-3)^7  = 7 (-2187)  = -15309

It depends upon what  her favorite integer might  be

Ex:

If it is 8  we  have   sqrt (8)  / sqrt (2) =  sqrt (4)  = 2

But......suppose it is 12

sqrt (12) /sqrt(2)   =  sqrt (6)    not an integer

See the following image  :

AB  = 10    BC  = 16

Since  FG is parallel to AB.....then triangle  GFC  is similar to  triangle  ABC

Therefore....

AB/BC  = GF/FC

10/16  = GF/ FC

5/8  = GF/FC

8/5 = FC / GF

Let  GF = the side  of the square  = s

So....

(8/5)s  = FC

Draw  altitude  AH

Triangle BHA is a right triangle

sin angle B  = 8/10  = 4/5

But triangle BEF  is also a right triangle  with  hypotenuse BF

Therefore  sin angle B   =  EF/ BF = s/BF

4/5 = s/BF

5/4  = BF/s

(5/4)s   =BF

But   BF  +  BC  = 16

Therefore

(8/5)s  + (5/4)s   =16

[( 32 + 25) / 20 ] s  = 16

(57/20)s  =16

s =  (20)(16)/57  =  320 / 57

What if the first card drawn was the 10 of Hearts? 

Then, there'd be only three 10's left in the deck. 

c=0;listforeach(n, d=(2, 3, 4, 5, 6, 6, 7, 8, 8, 9, 10, 10, 12, 12, 14, 15, 16, 18, 18, 20, 20, 21, 24, 24, 27, 28, 30, 30, 32, 35, 36, 40, 40, 42, 45, 48, 50, 54, 56, 60, 63, 70, 72, 80, 90), m=if(n%2==0, c=(c,n), 0);print c[1, count c-1],">>Total =", count c-1>>Total = 45 such products.

(2, 4, 6, 6, 8, 8, 10, 10, 12, 12, 14, 16, 18, 18, 20, 20, 24, 24, 28, 30, 30, 32, 36, 40, 40, 42, 48, 50, 54, 56, 60, 70, 72, 80, 90) >>Total = 35 EVEN numbers

Probability = 35 / 45 == 7 / 9