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pishuonlain [190]

3 years ago

11

£35.50 for the first day then £18.25 for each extra day the total cost of hiring the machine was £ 163.25 how many day dose Mart

in hire the machine for

1 answer:

BabaBlast [244]3 years ago

8 0

y = number of days
35.50 + 18.25 y = 163.25

18.25 y = 163.25 - 35.50

18.25y = 127.75

y = 127.75 / 18.25

y = 7

so Martin has hired the machine for 8 days

( first day paid 35.50), then 7 days for 18.25

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He lifted 100 kilograms and 132.5 kilograms. What was the combined weight of his lifts in grams?

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3 years ago

Davis made 4 clay pots on Friday,7 clay pots on Saturday and clay pots on Sunday. What is the mean number of pots he made each d

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3 years ago

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 $x\equiv147\pmod{64}\implies\boxed{x\equiv19\pmod{64}}$

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3 years ago

Use the given conditions to write an equation for the line in point-slope form and general form. Passing through (8 comma negat

miskamm [114]

Answer:

The answer to your question is y = -5x + 32 point-slope form

5x + y - 32 = 0 general form

Step-by-step explanation:

Data

(8, -8)

⊥ x - 5y - 6 = 0

Process

1.- Get the slope of the line given

x - 5y - 6 = 0

-5y = -x + 6

y = -x/-5 + 6/-5

y = x/5 - 6/5

slope = 1/5

slope of the new line -5, because the lines are perpendicular

2.- Get the equation of the new line

y - y1 = m(x - x1)

y + 8 = -5(x - 8)

y + 8 = -5x + 40

y = -5x + 40 - 8

y = -5x + 32 point-slope form

Equal to zero to find the general form

5x + y - 32 = 0 general form

5 0

3 years ago