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You have enough cash to pay 80% of the book price, so 0.8*price = $35

Hence price = $35/0.8. = $43.75  

2.

\(AA_5 = 5A + A = 6A \quad \forall \; A \in [0,4] \land A\in \mathbb Z\\ BB_7 = 7B + B = 8B \quad \forall \; B \in [0,6] \land B\in \mathbb Z\)

From this, we know that the required base-10 integer must be divisible by both 6 and 8.

Least common multiple of 6 and 8 = 24.

So the answer is 24.

Check: \(24_{10} = 44_5 = 33_7\)

1.

Obviously, \(n\geq 7\).

\(\quad 235236_n\\ =2n^5 + 3n^4 + 5n^3 + 2n^2 + 3n + 6\) 

This holds true for any \(n\geq 7\).

Then, clearly, \(2n^5+3n^4+5n^3+2n^2+3n\equiv 1\pmod7\).

Define \(P(n) = 2n^5+3n^4+5n^3+2n^2+3n-1\).

Consider all possible values of P(n) mod 7 when \(0 \leq n \leq 6, n\in \mathbb Z\).

\(P(0) \equiv 6\pmod 7\\ P(1) \equiv 0 \pmod 7\\ P(2) \equiv 4 \pmod 7\\ P(3) \equiv 1 \pmod 7\\ P(4) \equiv 1 \pmod 7\\ P(5) \equiv 1\pmod 7\\ P(6) \equiv 1 \pmod 7\\ \)

Generally, \(P(7k + 1) \equiv 0 \pmod 7 \quad\forall \;k\in \mathbb Z^+\).

All possible values of n are: 8,15,22,29,36,43,50,57,64,71,78,85,92,99.

So there are 14 possible values of n such that \(235236_n\) is divisible by 7.

\(\quad\dfrac{1}{4}+6-3\cdot 14\\ = \dfrac{-143}{4}\)

Thanks Melody for your patience and skills! 

any answer

x=||2a-3b||

\(7^2=4^2+4^2- 2*4*4*cos(A) \\ \Rightarrow cos(A)=-\dfrac{17}{32} \)

\(x^2 =8^2+12^2-2*8*12*cos(180^{\circ}-A )\\ x^2=208+192*cos(A) \)

\(x = \sqrt{106 }\)

You have not provided the actual question and  you do not even have enough knowledge to understand this fact.

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Your non-question.

The figure below is not drawn to scale. We are given that angle ADB is obtuse.
Which of the following statements must be true?

(i) BD (ii) CD 
Enter the letter of the multiple choice option below: 

A: (i) only 
B: (ii) only 
C: (i) and (ii) 
D: neither (i) nor (ii)

https://web2.0calc.com/questions/need-help-quick_4

Is that   AA in base 5    and BB in base 7 ?

           11 in base 5 = 6  in base 10          11 in base 7 = 8 in base 10

If you are looking for the SAME number, I think   24 in base 10 is represented by   44 in base 5    and 33 in base 7

I think this is correct

We can place the first pawn  in  any column of the top row  = 4 ways

For the second pawn....we can place it anywhere on the second row except in the column occupied by the first pawn = 3 ways

The third pawn can then be placed in any two positions on the third row....= 2 ways

And the last pawn, by default....only has 1 way it can be placed on the fourth row

So

4 x 3 x 2  x 1   =   24 ways

Here's my best effort....although there might be an easier method !!!

Wnen  = 0   we have the point (2, 2)

When t = 1  we have the point  (6, 3)

So...the slope of the given line is   (3 - 2) / (6 - 2) =  1/4

If we let the parallel line go through the origin....we have the line  y = (1/4)x  

This means that  every point on this line has the coordinates ( x, 1/4x)

This implies that

x + (1/4)x = 10

(5/4)x = 10

x = 8   and y = (1/4)x = 2

So....the required vector  is    < 8, 2 >

Oh, I see what you are saying. I thought the first condition with x+1 if floor x is prime was saying: If floor x is prime, then p(x) = floor x + 1.

That explains a lot. But again, the number can be infinitely close to 8, such as 7.99999..., but the number can never be 8, so the answer range is \(\boxed{[3,9)}\).

A  A(base 11)
[11^1 x 10] + [11^0 x 10] =120
120_5 =35 - the smallest base 10 number.

B B(base 12)
[12^1 x 11] + [12^0 x 11] =143
143_7 =80 - the smallest base 10 number.

Thank you!

IF you use the units, you can easily figure this out (as well as other , more comlicated equations)

density =  19.3  gm /cm^3     if you multiply this by volume 4.5 cm^3      the   'cm^3'   cancels out and you are left with only grams as units

19.3 gm/ cm^3  *  4.5 cm^3 = 86.85 gm

The directrix lies below the focus...so...this parabola turns upward....we can immediately rule out the last two answers

The vertex lies at (0, (-3 - 2) /2) =   (0, -5/2)

We have the form

4p ( y - k) =  (x -h)^2        where (h, k) is the vertex = (0, -5/2)

p is the distance between the focus and vertex = 1/2 units

So we have

4(1/2)(y - [- 5/2] ) = (x - 0)^2

2(y + 5/2) = x^2

2y + 5 = x ^2

2y = x^2 - 5      divide both sides by 2

y = (1/2)x^2 - 5/2

6 in  = 1/2 ft

And

 1 1/2 = 3/2 ft

So...the area is base * height =   (3/2) (1/2) =   (3/4)ft^2

A = (1, 1)

B = (2, 3)

C = (4, 0)

AB = sqrt [ (1 - 2)^2+ (3 - 1)^2 ] =  sqrt [ 5]

CA = sqrt  [(1 - 4)^2 + (1 - 0)^2 ] = sqrt [10]

BC = sqrt [ (2 - 4)^2 + (3 - 0)^2 ] = sqrt [ 13]

Using the Law of Cosines

cos (theta) =  ( [13 - 10 - 5]/   [ - 2sqrt (50) ])

arcos =  [ -2] / [-2 sqrt (50) ] = theta

arcos  = 1 /sqrt (50)  = theta

arcos  =1 / [5sqrt(2) ] = (1/5)(1/sqrt(2) )  =  (1/5)/sqrt (2) = theta

So

cos (theta) =   (1/5) / sqrt(2)

So.... x = 1/5

First, label the appropriate diagram of the parallelogram. If the diagonals are perpendicular to each other, then the sides of ABCD has to be 5.

use the equation for obtuse triangles

\(AB^2>25=3^2+4^2=AE^2+BE^2\)

all four angles around E, would be obtuse, which you can figure out to be impossible, so the side can't be longer than 5

If vectors are orthagonal, their dot product = 0

So

[ 4 * 3 ] + [ -7 * k] = 0

12 - 7k = 0

-7k = -12

k = 12/7

So

[2 * 3 ] + [12/7 * m] = 0

6 + (12/7)m = 0

(12/7)m = - 6

m = -6*7/12 = -7/2

Thank you both to heureka and Melody! I read the article, and used heureka's method to find another way to solve the problem.