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2 years ago

7

A box has dimensions of 19 inches long, 1.7 feet wide, and 6 inches high. What is the volume of the box? The formula for the vol

ume is V = l × w × h.

1 answer:

denis-greek [22]2 years ago

3 0

convert the feet to inches. then multiply all 3 numbers to get your answer

1.7 feet = 1.7 x 12 = 20.4 inches

19 x 20.4 x 6 = 2325.6 cubic inches

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Solve the following equations: (a) x^11=13 mod 35 (b) x^5=3 mod 64

tino4ka555 [31]

a.

$x^{11}=13\pmod{35}\implies\begin{cases}x^{11}\equiv13\equiv3\pmod5\\x^{11}\equiv13\equiv6\pmod7\end{cases}$

By Fermat's little theorem, we have

$x^{11}\equiv (x^5)^2x\equiv x^3\equiv3\pmod5$

$x^{11}\equiv x^7x^4\equiv x^5\equiv6\pmod 7$

5 and 7 are both prime, so $\varphi(5)=4$ and $\varphi(7)=6$ . By Euler's theorem, we get

$x^4\equiv1\pmod5\implies x\equiv3^{-1}\equiv2\pmod5$

$x^6\equiv1\pmod7\impleis x\equiv6^{-1}\equiv6\pmod7$

Now we can use the Chinese remainder theorem to solve for $x$ . Start with

$x=2\cdot7+5\cdot6$

Taken mod 5, the second term vanishes and $14\equiv4\pmod5$ . Multiply by the inverse of 4 mod 5 (4), then by 2.

$x=2\cdot7\cdot4\cdot2+5\cdot6$

Taken mod 7, the first term vanishes and $30\equiv2\pmod7$ . Multiply by the inverse of 2 mod 7 (4), then by 6.

$x=2\cdot7\cdot4\cdot2+5\cdot6\cdot4\cdot6$

$\implies x\equiv832\pmod{5\cdot7}\implies\boxed{x\equiv27\pmod{35}}$

b.

$x^5\equiv3\pmod{64}$

We have $\varphi(64)=32$ , so by Euler's theorem,

$x^{32}\equiv1\pmod{64}$

Now, raising both sides of the original congruence to the power of 6 gives

$x^{30}\equiv3^6\equiv729\equiv25\pmod{64}$

Then multiplying both sides by $x^2$ gives

$x^{32}\equiv25x^2\equiv1\pmod{64}$

so that $x^2$ is the inverse of 25 mod 64. To find this inverse, solve for $y$ in $25y\equiv1\pmod{64}$ . Using the Euclidean algorithm, we have

64 = 2*25 + 14

25 = 1*14 + 11

14 = 1*11 + 3

11 = 3*3 + 2

3 = 1*2 + 1

=> 1 = 9*64 - 23*25

so that $(-23)\cdot25\equiv1\pmod{64}\implies y=25^{-1}\equiv-23\equiv41\pmod{64}$ .

So we know

$25x^2\equiv1\pmod{64}\implies x^2\equiv41\pmod{64}$

Squaring both sides of this gives

$x^4\equiv1681\equiv17\pmod{64}$

and multiplying both sides by $x$ tells us

$x^5\equiv17x\equiv3\pmod{64}$

Use the Euclidean algorithm to solve for $x$ .

64 = 3*17 + 13

17 = 1*13 + 4

13 = 3*4 + 1

=> 1 = 4*64 - 15*17

so that $(-15)\cdot17\equiv1\pmod{64}\implies17^{-1}\equiv-15\equiv49\pmod{64}$ , and so $x\equiv147\pmod{64}\implies\boxed{x\equiv19\pmod{64}}$

5 0

2 years ago

Please help i will give you brainliest and every yhing

Lina20 [59]

first subtract the last equation from the first. this gives:-

-x + y = -8 .....................(1)

Then multiply the first equation by 2 and add it to the 2nd equations This gives

9x = 18

so x = 2

and from equation (1) y = -8 + 2 = -6

Substituting for x and y in the second equation

z = (24 -5(2) -12) / 2 = 1

Answer is choice b.

7 0

2 years ago

Find the product (3x-9)^2

Stells [14]

Here are the steps:

(3x-9)^2

(3x-9)(3x-9)

9x^2 - 27x - 27x +81

9x^2 - 54x + 81

good luck ;)

4 0

2 years ago

How would you do this?

s344n2d4d5 [400]

Answer:

aileenessayhelp.com

7 0

2 years ago

Read 2 more answers

Quadrilateral LMNO is similar to quadrilateral PQRS. Find the measure of side QR.

dalvyx [7]

Answer:

The measure of the side $QR$ is 8.8.

Step-by-step explanation:

Since $LMNO \sim PQRS$ , then $SP \propto OL$ , $RS \propto ON$ , $RQ \propto MN$ and $QP \propto LM$ . From figure we have the following relationship:

$k = \frac{OL}{SP} = \frac{ON}{RS} = \frac{MN}{QR} = \frac{ML}{QP}$ (1)

Where $k$ is the proportionality ratio.

If we know that $SP = 13, OL = 55, MN = 37$ , then the measure of side $QR$ is:

$k = \frac{OL}{SP}$ (1b)

$k = \frac{55}{13}$

$k = \frac{MN}{QR}$

$QR = \frac{MN}{k}$ (1c)

$QR = \frac{37}{\frac{55}{13} }$

$QR = 8.745$

The measure of the side $QR$ is 8.8.

4 0

2 years ago