a) The GCD of 27,720 and 58,212 is 1.
b) Integers r = 3 and s = -1 satisfy the equation 27720r + 58212s = GCD(27720, 58212).
How to find the greatest common divisor (GCD) of 27,720 and 58,212?(a) To find the greatest common divisor (GCD) of 27,720 and 58,212, we can use the Euclidean Algorithm.
Divide 58,212 by 27,720.
58,212 ÷ 27,720 = 2 remainder 2
Divide the previous divisor (27,720) by the remainder (2).
27,720 ÷ 2 = 13,860
Divide the previous remainder (2) by the new remainder (2).
2 ÷ 2 = 1
Since the remainder is now 1, the GCD is found.
Therefore, the greatest common divisor (GCD) of 27,720 and 58,212 is 1.
How to find integers r and s such that 27720r + 58212s = GCD(27720, 58212)?(b) To find integers r and s such that 27720r + 58212s = GCD(27720, 58212), we can use the Extended Euclidean Algorithm.
Using the Euclidean Algorithm from part (a), we can backtrack to find the coefficients r and s:
2 = 27,720 - 13,860
Substituting the values:
2 = 27,720 - (58,212 - 2 * 27,720)
Simplifying:
2 = 3 * 27,720 - 58,212
Comparing with the form 27720r + 58212s = GCD(27720, 58212):
r = 3
s = -1
Therefore, we substitute r = 3 and s = -1 into the equation 27720r + 58212s, it will result in the greatest common divisor (GCD) of 27720 and 58212.
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Let A and B be disjoint compact subspaces of a Hausdorff space X. Show that there exist disjoint open sets U and V, with ACU and BCV.
In Hausdorff-space "X", if A and B are disjoint "compact-subspaces", then there is disjoint "open-sets" U and V such that A is contained in U and B is contained in V, this is because by Hausdorff-Property, the existence of disjoint open neighborhoods for any two "distinct-points".
To prove the existence of disjoint "open-sets" U and V with A⊂U and B⊂V, where A and B are "compact-subspaces" of "Hausdorff-space" X,
Step (1) : A and B are disjoint compact-subspaces, we use Hausdorff property to find "open-sets" Uₐ and [tex]U_{b}[/tex] such that "A⊂Uₐ" and "B⊂[tex]U_{b}[/tex]", and "Uₐ∩[tex]U_{b}[/tex] = ∅". This can be done for every pair of points in A and B, respectively, because X is Hausdorff.
Step (2) : We consider, set U = ⋃ Uₐ, where "union" is taken over all of Uₐ for each-point in A. U is = union of "open-sets", hence open.
Step (3) : We consider set V = ⋃ [tex]U_{b}[/tex], where union is taken over for all [tex]U_{b}[/tex] for "every-point" in B. V is also a union of open-sets and so, open.
Step (4) : We claim that U and V are disjoint. Suppose there exists a point x in U∩V. Then x must be in Uₐ for some point a in A and also in [tex]U_{b}[/tex] for some point b in B. Since A and B are disjoint, a and b are different points. However, this contradicts the fact that Uₐ and [tex]U_{b}[/tex] are disjoint open sets.
Therefore, U and V are disjoint open sets with A⊂U and B⊂V.
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The given question is incomplete, the complete question is
Let A and B be disjoint compact subspaces of a Hausdorff space X. Show that there exist disjoint open sets U and V, with A⊂U and B⊂V.
What is the area of a rectangle with side lengths 2/5 feet and 4/6 feet?
Answer:
[tex]\frac{4}{15}[/tex] (4/15)
Step-by-step explanation:
[tex]\frac{4}{6}=\frac{2}{3}[/tex]
[tex]\frac{2}{5}*\frac{2}{3};[/tex]
1- Multiply the numerators:
[tex]2*2=4[/tex]
2- Multiply the denominators:
[tex]5*3=15[/tex]
3- Thus:
[tex]\frac{2}{5}*\frac{2}{3}= \frac{4}{15}[/tex]
Hope this helps ;)
what is 4836 divided by 9735829 plus 28369 times 284383?
Answer:
2.28871.0005^14
Step-by-step explanation:
Answer:
i got 8067661468.26
Step-by-step explanation:
A cylinder has a volume of 2,309.07 cubic cm and a height of 15 cm. What is the
radius of the cylinder? Use 3.14 for st in your calculations and round to the
nearest whole number.
cm
Answer:
7
Step-by-step explanation:
the answer I got was 7 hope this helped
Function f is a quadratic function passing through the points (-4,0),(0,–12) and (3,0). Function g is modeled by the graph. Over which interval are both functions negative?
Answer:
Open points on 1 and 3 with ray connecting them
Step-by-step explanation:
Because it's right.
Answer:
Step-by-step explanation:
What is the one shape of cross-section we could create by slicing the cylinder perpendicular to its base
Answer:
rectangle
Step-by-step explanation:
A rectangle represents a figure with four sides having equal opposite sides and the inside angle between the two adjacent sides is ninety degree.
When any plane is made to pass through the cylinder creating a cross section which is perpendicular to its base, it results a two-dimensional cross sectional shape of a rectangle.
Bacteria triples every 3 hours. If there are 450 bacteria at t=0
min how many after 200 min
Therefore, the number of bacteria after 200 minutes can be found by multiplying the initial number of bacteria (450) by the tripling factor (3) raised to the power of 2 (for the full cycles) and multiplying by the remaining fraction of the tripling factor for the partial cycle.
Since the bacteria triples every 3 hours, we can calculate the number of tripling cycles that have occurred in 200 minutes. Since 3 hours is equivalent to 180 minutes, there are 2 full cycles in 200 minutes. To calculate the remaining fraction of the tripling cycle, we divide the remaining time (20 minutes) by the length of a single cycle (180 minutes). The fraction is 20/180, which simplifies to 1/9.
Now, we can calculate the number of bacteria after 200 minutes. We start with the initial number of bacteria, which is 450, and multiply it by the tripling factor (3) raised to the power of the number of full cycles (2). This accounts for the full cycles. Then, we multiply this result by the remaining fraction of the tripling factor (1/9) to account for the partial cycle.
Therefore, the number of bacteria after 200 minutes can be calculated as follows:
Number of bacteria = 450 * (3^2) * (1/9) = 450 * 9 * (1/9) = 450
Hence, after 200 minutes, there will still be 450 bacteria.
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(22) + (3x) = 4
solve for x
Answer:
-6
Step-by-step explanation:
Step One: The goal is to isolate the x. So first, we would do 4-22, which is -18. The equation is: 3x=-18.
Step Two: Lastly, we need to divide by three to completely isolate the x. This is -6.
Consider the derivation of the quadratic formula below. What is the missing radicand
in Step 6?
Answer:
[tex]\frac{b^2 - 4ac}{4a^2}[/tex]
Step-by-step explanation:
Given
See attachment for complete question
Required:
Complete step 6
At step 5, we have:
[tex](x + \frac{b}{2a})^2 = \frac{b^2}{4a^2} - \frac{4ac}{4a^2}[/tex]
Take LCM
[tex](x + \frac{b}{2a})^2 = \frac{b^2 - 4ac}{4a^2}[/tex]
Take square roots of both sides to get step 6
[tex]x + \frac{b}{2a} = \±\sqrt{\frac{b^2 - 4ac}{4a^2}}[/tex]
Hence, the missing radicand is: [tex]\frac{b^2 - 4ac}{4a^2}[/tex]
Help with question 3 please I’ll give brainlest
Answer: b
Step-by-step explanation: When you rotate it around the axis (think of it as a pole) it will become a cylinder with radius 5
a) The difference of the age of two sisters is 5 years and the product of their age is
24. Find the age of the two sisters.
Answer:
3 years old and 8 years old
Step-by-step explanation:
8-3=5
3x8=24
plz mark me as brainliest
Identify the like terms in the expression 4x+5-x^3-3x
solve for x to the nearest
Answer:
in right angled triangleBCD
BC=√{DC²-BC²)=√{10²-6²)=8
again in right angled triangle ABC
AB=√(BC²-AC²)
x=√(8²-7²)=3.87
Jim , a meteorologist for local television XYZ, would like to report the rainfall. The following are the rainfall measurements (in inches) for to-day’s date for 14 randomly chosen past years. The data are as follows: 0.47 0.27 0.13 0.54 0.08 1.05 0.34 0.26 0.42 0.17 0.50 0.86 0.01 2.5 NOTE: answers should be given to 2 decimal places REQUIRED 1. Find the RANGE of the data provided and what measure does this show 2. Calculate the sample variance . Why is variance not written in? 3. Calculate the sample standard deviation. What information does this measure give to the users? Why is it necessary to compute standard deviation instead of stopping at the variance calculation? 4. Find the COEFFICIENT OF VARIATION 5. Use Z-score to find if there is any outliers in the series
Where the above is given,
Range -2.49 inchesSample variance - 0.5599Sample standard deviation - 0.7483Coefficient of variation - 7497.32% Z14 = 3.3999, which is greater than 2, suggesting that the data point 2.50 inches may be considered an outlier based on the Z-Score criterion.What is the explanation for the above ?Range
Range = Maximum value - Minimum value
Range = 2.50 - 0.01
= 2.49 inches
Sample Variance
Step 1 - Calculate the mean (average) of the data.
Mean = (0.47 + 0.27 +0.13 + 0.54 + 0.08 + 1.05 + 0.34 + 0.26 + 0.42 + 0.17 + 0.50 + 0.86 + 0.01 + 2.5) / 14
= 1.36 /14
= 0.0971
Step 2 - Calculate the squared deviation from the mean for each data point.
Deviation from mean = Data point - Mean
Squared deviation = (Deviation from mean)²
Sum of squared deviations = (0.3739 + 0.0091 + 0.7307 + 0.1638 + 0.9469 + 0.104 + 0.0145 + 0.0081 + 0.0971 + 0.3037 + 0.0601 + 0.2601 + 1.0679 + 3.1391)
= 7.2799
Sample Variance = Sum of squared deviations / (n - 1)
= 7.2799/ (14 - 1)
= 0.5599
Sample Standard Deviation:
Sample Standard Deviation = √(Sample Variance)
= √0.5599
= 0.7483
Coefficient of Variation:
Coefficient of Variation = (Sample Standard Deviation / Mean) * 100
= (7.2799/ 0.0971) * 100
= 7497.32
Z-Scores for each data point
Z1= (0.47 - 0.0971)/ 0.7212= 0.5318
Z2= (0.27 - 0.0971)/ 0.7212= 0.9983
Z3= (0.13 - 0.0971)/ 0.7212= 0.0456
Z4= (0.54 - 0.0971)/ 0.7212= 0.6189
Z5= (0.08 - 0.0971)/ 0.7212= -0.0251
Z6= (1.05 - 0.0971)/ 0.7212= 1.2914
Z7= (0.34 - 0.0971)/ 0.7212= 0.4574
Z8= (0.26 - 0.0971)/ 0.7212= 0.2251
Z9= (0.42 - 0.0971)/ 0.7212= 0.4969
Z10= (0.17 - 0.0971)/ 0.7212= 0.1002
Z11= (0.50 - 0.0971)/ 0.7212= 0.6506
Z12= (0.86 - 0.0971)/ 0.7212= 1.0242
Z13= (0.01 - 0.0971)/ 0.7212= -0.1208
Z14= (2.50 - 0.0971)/ 0.7212= 3.3999
To determine if there are any outliers, we can compare the absolute values of the Z-Scores to a certain threshold , commonly considered as 2.
If the absolute value of a Z-Score is greater than 2, it indicates that the corresponding data point is an outlier.
In this case, Z14 = 3.3999, which is greater than 2 , suggesting that the data point 2.50 inches may be considered an outlier based on the Z-Score criterion.
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A fresh food distributor receives orders from 100 customers daily. Assume that the quantities ordered by customers, in kg, are independent continuous random variables uniformly distributed over the interval (0, 9). Assuming that the distributor only has the capacity to ship 477 kg of products daily, calculate the probability that all orders are fulfilled on a day chosen at random. Indicate the result to at least four decimal places.
The probability that all orders are fulfilled on a day chosen at random is approximately 1. Answer: 1.0000 (rounded off to at least four decimal places).
The quantities ordered by customers are independent continuous random variables, and they are uniformly distributed over the interval (0, 9).
The fresh food distributor only has the capacity to ship 477 kg of products daily, and the distributor receives orders from 100 customers daily.
The probability that all orders are fulfilled on a day chosen at random is given by;P(all orders fulfilled) = P(X1 + X2 + ... + X100 < 477)
where X is the quantity ordered by each customer. Since X is a continuous random variable, we can use the probability density function of a uniform distribution to calculate the probability density function of X as;f(x) = 1/9, 0 < x < 9
Hence, the probability that all orders are fulfilled on a day chosen at random is given by;
P(all orders fulfilled) = P(X1 + X2 + ... + X100 < 477)= P[(X1/9) + (X2/9) + ... + (X100/9) < (477/9)]= P[U < (53 + 1/3)], where U ~ Uniform(0, 1)
Now, using the central limit theorem, we can approximate the distribution of U by a normal distribution with mean μ = 1/2 and variance σ^2 = 1/12 such that;Z = (U - μ) / σ ~ N(0, 1)
Hence, P[U < (53 + 1/3)] = P[Z < (53 + 1/3 - μ) / σ]= P[Z < (53 + 1/3 - 1/2) / sqrt(1/12)]≈ P[Z < 9.6067]≈ 1
Thus, the probability that all orders are fulfilled on a day chosen at random is approximately 1. Answer: 1.0000 (rounded off to at least four decimal places).
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The odds against a certain football team winning the championship are 70 : 1. a) Determine the probability that the team wins the championship. b) Determine the probability that the team does not win the championship.
a) The probability that the team wins the championship is approximately 0.0141 or 1.41%. b) The probability that the team does not win the championship is approximately 0.9859 or 98.59%.
a) To determine the probability that the team wins the championship, we need to convert the odds against winning into a probability.
The odds against winning are given as 70:1. This means that for every 70 unfavorable outcomes (losing), there is 1 favorable outcome (winning).
To calculate the probability of winning, we divide 1 by the sum of the favorable and unfavorable outcomes:
Probability of winning = 1 / (70 + 1) = 1 / 71 ≈ 0.0141 (or 1.41%)
Therefore, the probability that the team wins the championship is approximately 0.0141 or 1.41%.
b) The probability of not winning the championship is equal to 1 minus the probability of winning:
Probability of not winning = 1 - 0.0141 = 0.9859 (or 98.59%)
Therefore, the probability that the team does not win the championship is approximately 0.9859 or 98.59%.
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sales tax: 68% shirts: $35 pants: $27 shoes: $44 what is the total cost
Simplify 4(3n + 2b + k)
a. 12nbk
b. 12n + 2bk
c. 12n + 8b + 4k
d. 24bkn
Answer:
C
Step-by-step explanation:
4(3n+2b+k)
12n+8b+4k
So it would be C.
---
hope it helps
Part A
An economist has measured people's annual salary (in thousands of dollars) and their years of relevant job experience, thinking that a linear relationship between them might exist.
Let the proposed regression relationship between Salary and experience be as follows: E(Salary) = beta subscript 0 space plus space beta subscript 1 space cross times Years of Experience
and assume the output from running the regression is as follows:
Call:
lm(formula = Salary ~ Year, data = Income)
Residuals:
Min 1Q Median 3Q Max
-53.650 -20.256 0.127 18.423 65.596
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 31.8387 8.5565 3.721 0.00033***
Years 2.8205 0.3302 8.543 1.74e-13 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 25.98 on 98 degrees of freedom
Multiple R-squared: 0.4268, Adjusted R-squared: 0.421
F-statistic: 72.98 on 1 and 98 DF, p-value: 1.737e-13
---
Residual standard error: 8.044 on 445 degrees of freedom
Multiple R-squared: 0.6914, Adjusted R-squared: 0.6886
F-statistic: 249.2 on 4 and 445 DF, p-value: < 2.2e-16
If we wished to conduct a hypothesis test as to whether there is a linear relationship between salary and years of experience, what are the correct null and alternate hypotheses?
Answers:
a.
H subscript 0 : space beta subscript 0 space equals space 0 H subscript 1 : space beta subscript 0 greater than space 0
b.
H subscript 0 : space beta subscript 0 space equals space 0 H subscript 1 : space beta subscript 0 space end subscript not equal to space 0
c.
H subscript 0 space : thin space beta subscript 1 space equals space 0 H subscript 1 : space beta subscript 1 space end subscript space not equal to space 0
d.
H subscript 0 : space beta subscript 1 space equals space 0 H subscript 1 : space beta subscript 1 space greater than space 0
Part B
Using the output in Q1, what is the correct p-value for the test in Q1?
Answers:
a.
0.00033
b.
0.000000000000174
c.
1.74e-13
d.
0.00000393
Part C
What is the fitted regression model from this output in Q1?
Answers:
a.
E(Salary) = 31.8387 + 2.8205 x Years of Experience
b.
E( Years of Experience ) = 2.8205 + 31.8387 x Salary
c.
E( Years of Experience ) = 31.8387 + 2.8205 x Salary
d.
E(Salary) = 2.8205 + 31.8387 x Years of Experience
Part D
Which of the following is a correct statement regarding r squared ?
Answers:
a.
r squared space equals space 0.4268 meaning that Years of Experience explains 42.68 percent sign of the variability in Salary.
b.
r squared space equals space 0.00033 meaning that Years of Experience explains 0.033 percent sign of the variability in Salary.
c.
r squared space equals space 0.00033 and because 0.00033 space less than space 0.05 we reject H subscript 0 and accept H subscript 1 at the 5% level of significance, ie we conclude there is a significant linear relationship between Salary and Years of Experience.
d.
r squared space equals space 0.4268 and because 0.4268 space greater than space 0.05 we do not reject H subscript 0 at the 5% level of significance, ie we conclude there is no significant linear relationship between Salary and Years of Experience.
The correct statement regarding r squared is:
r squared equals 0.4268 meaning that Years of Experience explains 42.68 percent of the variability in Salary.
Part A: The correct null and alternate hypotheses are:
H₀: β₁=0;
H₁: β₁≠0.
Part B: The correct p-value for the test in Q1 is 1.74e-13.
Part C: The fitted regression model from this output in Q1 is:
E(Salary) = 31.8387 + 2.8205 x Years of Experience.
Part D: The correct statement regarding r squared is:
r squared equals 0.4268 meaning that Years of Experience explains 42.68 percent of the variability in Salary.
Explanation: The output shows a multiple linear regression model:
Salary=β0+β1x
Years of Experience + ϵ.β0 is the intercept and represents the expected mean salary for an individual with 0 years of experience.
β1 is the slope and represents the expected change in salary due to one year increase in experience.
ϵ is the error term (deviation from the expected salary).
The correct null and alternate hypotheses are:
H₀: β₁=0 (there is no linear relationship between salary and years of experience).
H₁: β₁≠0 (there is a linear relationship between salary and years of experience).
The correct p-value for the test in Q1 is 1.74e-13, which is much smaller than the significance level of 0.05.
Thus, we reject the null hypothesis and conclude that there is sufficient evidence to suggest that there is a linear relationship between salary and years of experience.
The fitted regression model from this output in Q1 is:
E(Salary) = 31.8387 + 2.8205 x Years of Experience.
The coefficient of determination, or R-squared, is a statistical measure that shows how well the regression model fits the observed data.
It is the proportion of the variance in the dependent variable that is explained by the independent variable(s).
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2/6, 5/12, 3/7, and 4/10. List least to most
the acceleration of an oscillator undergoing simple harmonic motion is described by the equation ax(t)=−(18m/s2)cos(33t) , where the time t is measured in seconds.
The equation [tex]ax(t) = -(18 m/s^2)cos(33t)[/tex] describes the acceleration of an oscillator. The acceleration varies sinusoidally with time, following a cosine function, and has a maximum value of [tex]-18 m/s^2[/tex].
The given equation [tex]ax(t) = -(18 m/s^2)cos(33t)[/tex] represents the acceleration of an oscillator undergoing simple harmonic motion. In this equation, t represents time measured in seconds.
The term cos(33t) indicates that acceleration varies sinusoidally with time. The cosine function has a period of 2π, meaning it completes one full cycle over the interval [0, 2π]. The coefficient 33 in front of t determines the frequency of oscillation. In this case, the oscillator completes approximately 33 cycles per second.
The negative sign indicates that the acceleration is directed opposite to the displacement of the oscillator. As the oscillator moves in one direction, the acceleration pulls it back in the opposite direction, causing it to oscillate around a stable equilibrium position.
The maximum acceleration is given by [tex]-18 m/s^2[/tex], which represents the amplitude of the oscillation. The acceleration varies between [tex]-18 m/s^2[/tex] and [tex]18 m/s^2[/tex], with the maximum magnitude occurring when the cosine function is at its peak value of 1 or -1.
Overall, the equation describes the acceleration of an oscillator undergoing simple harmonic motion, providing information about its amplitude, frequency, and direction of motion.
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Find the distance between the points (–8,10) and (4,10).
Answer:
12
Step-by-step explanation:
√(x2 - x1)² + (y2 - y1)²
√[4 - (-8)]² + (10 - 10)²
√(12)² + (0)²
√144 + 0
√144
=12
Which expression is equivalent to 1.5a + 2.4 (a + 0.5b) - 0.2b?
a. 3.9a + b
b. 2.5a + 0.7b
c. 3.9a + 0.3b
d. 2.5a + 0.3b + 2.4
The answer is A, 3.9a + b
Step-by-step explanation:
youre welcome
2. How does the graph of the following function compare to the quadratic parent function? * (1 Point) 8 (x) = x2 + 5 Moves up 5 Moves down 5 Moves to the left 5 Moves to the right 5
Answer:
b
Step-by-step explanation:
oi did the quiz f. 6373737
In a certain chemical, the ratio of zinc to copper is 3 to 14. Ajar of the chemical contains
546 grams of copper. How many grams of zinc does it contain?
It contains grams of zinc
Answer:
Given -
In a certain chemical, the ratio of zinc to copper is 4 to 13.
The chemical contains 546 grams of copper.
Prove -
How many grams of zinc does it chemical contain.
Answer
suppose that scalar multiple of the zinc and copper be y .
As given
In a certain chemical, the ratio of zinc to copper is 4 to 13.
The chemical contains 546 grams of copper.
Than equation is
13y = 546
13y = 546
y = 42
Zinc contain in the certain chemical = 4y
= 4 42
= 168 grams
Therefore 168 grams of zinc contain in a certain chemical .Step-by-step explanation:
If Y is inversely proportional to x and y=4 when x= 100, what is the value of y when x=250
Answer:
y=10
Step-by-step explanation:
[tex]\frac{4}{100}[/tex]=[tex]\frac{y}{250}[/tex]
cross-multiply, 4*250=100y
isolate the variable and solve for y, 1000=100y
divide 100 on both sides, 10=y
A guy wire supporting a radio tower is attached to the tower 128 feet above the ground. The wire makes a 45 degree angle with the ground. How long is the guy wire
Answer:
181 feet
Step-by-step explanation:
From the the diagram attached ,
Sinθ = a/b..................... Equation 1
Where θ = angle to the horizontal, a = Height of the tower, b = length of the wire.
make b the subject of the equation
b = a/sinθ................. Equation 2
Given: a = 128 feet, θ = 45°
Substitute these values into equation 2
b = 128/sin45°
b = 128/0.7071
b = 181 feet
b = 181 feet
43 - u = 69
how much is u?
Answer:
69-43= 26
therefore "U" is worth 26 U=26
Step-by-step explanation:
Answer:
its 26, because whaen u subtract 43 from 69 u will get 26
Compute in the ancient Egyptian way: (b) 55÷6 (a) 26 ÷ 20 (c) 71 21 ÷ (d) 25 18 (e) 52 ÷ 68 (f) 13 36
Ancient Egyptian way of computation, division was performed using a method called "repeated subtraction." (a) 26 ÷ 20 = 1 remainder 6.
(b) 55 ÷ 6 = 9 remainder 19.
(c) 71 21 ÷ = 50 remainder 29.
(d) 25 18 ÷ = 7.
(e) 52 ÷ 68 = 0 remainder 52.
(f) 13 36 ÷ = 0 remainder -23.
In the ancient Egyptian way of computation, division was performed using a method called "repeated subtraction." Here's how it would be applied to the given divisions:
(a) 26 ÷ 20:
To divide 26 by 20, we repeatedly subtract 20 from 26 until we cannot subtract anymore. The number of times we subtract is the quotient.
26 - 20 = 6
6 - 20 = -14 (cannot subtract anymore)
Therefore, 26 ÷ 20 = 1 remainder 6.
(b) 55 ÷ 6:
Using the same method, we repeatedly subtract 6 from 55 until we cannot subtract anymore.
55 - 6 = 49
49 - 6 = 43
43 - 6 = 37
37 - 6 = 31
31 - 6 = 25
25 - 6 = 19 (cannot subtract anymore)
Therefore, 55 ÷ 6 = 9 remainder 19.
(c) 71 21 ÷:
To divide 71 21 by a number, we first convert it to a whole number by multiplying the fraction part by the denominator.
71 21 = 71 + (21/100) = 71 + 21/100
Now, we can perform division using repeated subtraction.
71 - 21 = 50
50 - 21 = 29 (cannot subtract anymore)
Therefore, 71 21 ÷ = 50 remainder 29.
(d) 25 18 ÷:
Similar to the previous case, we convert 25 18 to a whole number.
25 18 = 25 + (18/100) = 25 + 18/100
Performing division:
25 - 18 = 7
Therefore, 25 18 ÷ = 7.
(e) 52 ÷ 68:
Since 52 is smaller than 68, the quotient is 0.
Therefore, 52 ÷ 68 = 0 remainder 52.
(f) 13 36 ÷:
Converting to a whole number:
13 36 = 13 + (36/100) = 13 + 36/100
Performing division:
13 - 36 = -23 (cannot subtract anymore)
Therefore, 13 36 ÷ = 0 remainder -23.
Please note that the ancient Egyptian method of division is not as efficient as modern division methods and may not produce exact decimal results.
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Express (–1+iV3) and (-1 – iV3) in the exponential form to show that:
2nnt (-1+iV3)n +(-1 – iV3)n = 2n+1cos 3
The proof of 2ⁿ (-1 + i√3)ⁿ + (-1 - i√3)ⁿ can be expressed as 2ⁿ⁺¹cos(πn/3) is proved below.
To express (-1 + i√3) and (-1 - i√3) in exponential form, we can use Euler's formula, which states that [tex]e^{(i\theta)[/tex] = cos(θ) + isin(θ).
Let's start with (-1 + i√3):
(-1 + i√3) = 2 x (cos(π) + i x sin(π/3))
Now, let's simplify (-1 - i√3):
(-1 - i√3) = 2 (cos(π) - isin(π/3))
Therefore, we have:
(-1 + i√3) = 2 e^(iπ/3)
(-1 - i√3) = 2 e^(-iπ/3)
Now, let's substitute these exponential forms into the expression:
2ⁿ (-1 + i√3)^n + (-1 - i√3)^n
= 2ⁿ(2 e^(iπ/3))^n + (2 e^(-iπ/3))^n
= 2ⁿ⁺¹ e^(iπn/3) + 2^(n+1) e^(-iπn/3)
Using Euler's formula again, we know that [tex]e^{(i\theta)} + e^{(-i\theta)[/tex] = 2cos(θ).
Therefore, we can rewrite the expression as:
2ⁿ⁺¹ (cos(πn/3) + cos(-πn/3))
= 2ⁿ⁺¹(cos(πn/3) + cos(πn/3))
= 2ⁿ⁺¹ 2 cos(πn/3)
= 2ⁿ⁺¹cos(πn/3)
So, we have shown that 2ⁿ (-1 + i√3)ⁿ + (-1 - i√3)ⁿ can be expressed as 2ⁿ⁺¹cos(πn/3).
Learn more about Exponential form here:
https://brainly.com/question/29166310
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