let p be prime. find all x ∈ zpsuch that x2 ≡ 1 (mod p). show your work. what is special about p = 2?
Answers
To find all x ∈ zp such that x^2 ≡ 1 (mod p), we can use the fact that (−1)^2 ≡ 1 (mod p) and (1)^2 ≡ 1 (mod p). Therefore, x ≡ 1 (mod p) and x ≡ −1 (mod p) are the only solutions.
If p = 2, then zp = {0, 1} and we can check that 1^2 ≡ 1 (mod 2) and (−1)^2 ≡ 1 (mod 2), which means that the same solutions as above apply. However, in this case, −1 ≡ 1 (mod 2), so the two solutions collapse into a single solution x ≡ 1 (mod 2). This is the special property of p = 2.
Given that p is prime, we want to find all x ∈ Zp such that x^2 ≡ 1 (mod p).
By definition, x^2 ≡ 1 (mod p) if and only if p divides (x^2 - 1). Now, we can rewrite x^2 - 1 as (x - 1)(x + 1). Since p is prime, it must either divide (x - 1) or (x + 1).
Case 1: p divides (x - 1)
In this case, x ≡ 1 (mod p).
Case 2: p divides (x + 1)
In this case, x ≡ -1 (mod p).
So, the possible values for x are 1 and -1 (mod p). For any prime p, these are the only two solutions.
For the special case when p = 2, we have x^2 ≡ 1 (mod 2). The solutions are x ≡ 1 (mod 2) and x ≡ -1 (mod 2). Since -1 is congruent to 1 (mod 2), there is only one unique solution for x in this case, which is x ≡ 1 (mod 2).
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Related Questions
Joseph's lunch at a restaurant costs $13.00 without tax. He leaves the waiter a tip of 17% of the cost of the lunch, without tax. What is the total cost of the lunch, including the tip
Answers
Answer:
The total cost of the lunch including the tip is $15.21.
Step-by-step explanation:
To find the total cost of the lunch including the tip, first, you need to calculate 17% of the price of the lunch as the statement indicates that that is the tip Joseph leaves the waiter:
$13*17%=$2.21
Now, you need to add that amount to the price of the lunch:
$13+$2.21=$15.21
According to this, the answer is that the total cost of the lunch including the tip is $15.21.
I will give you a Brainliest
Answers
Answer:
B isa the answer for me it might be wrong for you or switch them
Step-by-step explanation:
EFGH is translated 6 units to the left and 3 units up.
What are the coordinates of F’?
Answers
The coordinates of F' would be (2, 4). This is because F was translated 6 units to the left and 3 units up. To calculate the coordinates of F', you must subtract 6 from the x-coordinate of F and add 3 to the y-coordinate of F. Therefore, if F was (8, 1), then F' would be (2, 4).
F = (8, 1)
F' = (8 - 6, 1 + 3)
F' = (2, 4)
The coordinates of F' can be found by translating F 6 units to the left and 3 units up. To calculate the coordinates of F', you must subtract 6 from the x-coordinate of F and add 3 to the y-coordinate of F. For example, if F is located at (8, 1), the coordinates of F' would be (2, 4). This is because 8 - 6 = 2 and 1 + 3 = 4. Therefore, translating F 6 units to the left and 3 units up would result in the coordinates of F' being (2, 4). This principle applies to any initial coordinates, so you can use this method to calculate the coordinates of F' for any starting point.
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Solve each equation a x - b x / x + C equals D if a cannot equal to B + d
Answers
Answer:
x=dc/a-b-d
I hope this helps
A bike rental company charges $12.75 for the bike and $0.50 for each mile
Hidden. The total charge for one day's rental is $28.75. How many miles did
Answers
Answer:
32 miles
Step-by-step explanation:
this is because you first subtract 12.75 from 28.75 and this will leave you with 16.00. second you will divide 16.00 by .50 and this will leave you with 32 miles
After Sergio is done visiting the Empire State Building and walking around he will go yo: the Statue of Liberty (6.1 miles from the Empire State). the hotel (21.1 miles from the Statue of Liberty). How much would Sergio have spent on transportation using UberRUSH at the end of the day?
Answers
Supposing a linear function for the UberRUSH cost, we find that Sergio would have spent $23.6 in transportation.
The linear function for the UberRUSH cost has the following format:
\(C(m) = C(0) + am\)
In which:
C(0) is the fixed fee.a is the cost per mile.Supposing a fixed fee of $10 plus $0.5 per mile, we have that \(C(0) = 10, a = 0.5\), and:
\(C(m) = 10 + 0.5m\)
Sergio will use UberRUSH for 6.1 + 21.1 = 27.2 miles, thus, the cost is of:
\(C(27.2) = 10 + 0.5(27.2) = 23.6\)
Sergio would have spent $23.6 in transportation.
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Sony's utility function is U(q
1
,q
2
)=q
1
+Aq
1
a
q
2
b
+q
2
. The letters A,a,b are all positive constants. a) Find the marginal utility functions U
1
,U
2
of the two goods. b) Find the MRS. (Dorrit worry about reducing the math expression, it's not simplifiable in this example.)
Answers
The marginal utility functions for the given utility function are \(U_{1} = 1 + Aaq_{1} ^{(a-1)}q_{2}^{b}\) and \(U_{2} = Abq_{1} ^{a} q_{2}^{(b-1)}+ 1\). The MRS is equal to the ratio of the marginal utilities, or MRS = U₁/U₂ = \(1 + Aaq_{1} ^{(a-1)}q_{2}^{b}\)/ \((Abq_{1}^aq_{2}^{(b-1)} + 1)\).
a) The marginal utility functions can be obtained by taking the partial derivatives of the utility function with respect to each good. For the given utility function U(q₁, q₂) = \(q_{1} + Aq_{1}^{(a)}q_{2}^{(b)} + q_{2}\) the marginal utility of good 1 (U₁) is equal to\(1 + Aaq_{1}^{(a-1)}q_{2}^{(b)}\), and the marginal utility of good 2 (U₂) is equal to \(Abq_{1}^{(a)}q_{2}^{(b-1)} + 1\).
b) The marginal rate of substitution (MRS) represents the rate at which a consumer is willing to exchange one good for another while maintaining the same level of utility. It is defined as the ratio of the marginal utilities of the two goods. In this case, the MRS can be calculated as MRS = U₁/U₂, which gives \(\frac{(1 + Aaq_{1}^{(a-1)}q_{2}^{(b)})}{(Abq_{1}^{(a)}q_{2}^{(b-1)} + 1)}\).
The explanation above summarizes the process of obtaining the marginal utility functions and the MRS for the given utility function. The utility function is differentiated with respect to each good to find the marginal utilities. The MRS is then calculated as the ratio of the marginal utilities. The specific expressions for the marginal utilities and the MRS are provided based on the given utility function.
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Two African tigers were
weighed. One weighs 5x + 17
pounds and the two together
weigh 16x + 3 pounds. How
much does the other tiger
weigh alone?
Answers
Answer:
Step-by-step explanation:
The weight of the other tiger is 11x - 14.
The total weight of the two tigers is 16x + 3. Subtract the weight of one (5x + 17) from this total; the result is the weight of the other tiger:
16x + 3
- ( 5x + 17)
----------------
11x - 14
The weight of the other tiger is 11x - 14.
The graph of function f is shown. Function g is represented by the table. x -2 -1 0 1 2 g(x) -32 -16 -8 -4 -2 Which statement correctly compares the two functions on the interval [0, 2]? A. Function f is increasing, and function g is decreasing. B. Both functions are increasing, but function f is increasing faster. C. Both functions are increasing, but function g is increasing faster. D. Both functions are increasing at the same rate.
Answers
Answer:
Both functions are increasing, but function g is increasing faster
Step-by-step explanation:
Correct on a plato test. Have a great day!
Answer:
B. fashoooooooo
Step-by-step explanation:
Both functions are increasing, but function f is increasing faster.
is the standard deviation of the numbers x, y, and z equal to the standard deviation of 10, 15, and 20 ?
Answers
This matches the placement of 10, 15, and 20 and hence the standard deviation will be the same in the two cases.
What is the standard deviation?
The standard deviation in statistics is a measure of the amount of variation or dispersion in a set of values. A low standard deviation indicates that the values are close to the set's mean, whereas a high standard deviation indicates that the values are spread out over a wider range.
Standard deviation determines how far numbers deviate from the mean. The Standard deviation of the two sets will be the same if the numbers from the respective means are placed in the same order.
This is what 10, 15, and 20 will be on a number line
10_ _ _ _15 _ _ _ _20
(15 is the mean and 10 and 20 are 5 steps away from the mean)
i) Z - X = 10
This is what Z and X will be on the number line
X _ _ _ _ _ Z
ii) Z - Y = 5
This is what Z and Y will on the number line.
Y_ _ _ _ _ Z
Together, their relative placement on the number line:
X _ _ _ _ _ Y _ _ _ _ _ Z
This matches the placement of 10, 15, and 20 and hence the standard deviation will be the same in the two cases.
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The goal of a benefit concert is to collect 50,000 cans of food. Food is collected at each entrance to the concert.
Entrance A, 15,682 cans
Entrance B, 18,940 cans
Entrance C, 16,114 cans
Entrance D, 12,468 cans
Did they collect more or fewer cans than desired? How many more or fewer?
A.
13,204 more
B.
3,204 more
C.
13,204 fewer
D.
3,204 fewer
Answers
If you take all those and add it together and subtract it, it comes out to 13,204.
anemia for the study of jordanian children in exercise 4, the sample mean hemoglobin level was 11.3 g/dl and the sample standard deviation was 1.6 g/dl. a significance test yields a p-value of 0.0016. (a) interpret the p-value in context. (b) what conclusion would you make if a 0.05? a 0.01? justify your answer.
Answers
(a) The interpret the p-value in context of 0.0016 indicates that the probability of obtaining a sample mean hemoglobin level of 11.3 g/dl or lower.
(b) The conclusion would make if a 0.05 is rejected because the p-value (0.0016) is less than the significance level.
If using a significance level of 0.01, the null hypothesis would still be rejected because the p-value is less than 0.01.
Anemia Study Results(a) The p-value of 0.0016 indicates that the probability of obtaining a sample mean hemoglobin level of 11.3 g/dl or lower, assuming that the true population mean is 12 g/dl (or higher), is only 0.0016. This suggests that the observed difference between the sample mean and the hypothesized population mean is statistically significant and unlikely to be due to chance alone.
(b) If using a significance level of 0.05, the null hypothesis (that the true population mean is 12 g/dl or higher) would be rejected because the p-value (0.0016) is less than the significance level. Therefore, it can be concluded that the sample provides evidence that the true population mean hemoglobin level is lower than 12 g/dl.
If using a significance level of 0.01, the null hypothesis would still be rejected because the p-value is less than 0.01. In this case, the evidence is even stronger, and the conclusion that the true population mean is lower than 12 g/dl would be even more robust.
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What is the solution of log (2 t + 4) = log (14 minus 3 t)?
Answers
Answer:
-2
Step-by-step explanation:
Answer:
B) -2
Step-by-step explanation:
Just took the test and got it correct
Drag the cards to create a two-digit number that matches these rules. Note: Not all cards will be used 6 is one of the digits 20 is the nearest ten when rounding the cards 0 1 2 6
Answers
Answer:
As i understand this:
We have cards with the values:
0, 1, 2 and 6.
We want to create a two digit number such that:
Not all the cards are used.
6 is one digit of our number.
When we round to the nearest ten, we get 20.
Now, first remember how rounding works.
If we have a number like ab, where a and b are single digits (a is in the tens place and b is in the units place), then:
if b is 5 or more, we round up
if b is less than 5, we round down.
Now in this case we want to have a digit equal 6, if we use this in the units place, then when rounding, we will round up.
Then the other digit can be 1, such that we use the cards:
1 and 6.
then our number is 16:
When rounding, 6 > 5, then we round up to 20.
So this number meets all the conditions.
z scores that fall below the mean are ______. a. 0 b. negative c. either positive or negative d. positive
Answers
The correct answer to the question "z scores that fall below the mean are" is "negative". Z-scores are measures of relative standing that can be used to make comparisons among the scores in the distribution by transforming raw data into standardized units.
The formula for z-score is:z = (X-μ)/σz-scores can be negative, zero, or positive numbers. A negative z-score tells us that the corresponding score is below the mean and vice versa.In addition to this, the mean of all z-scores is always zero. If the mean of all z-scores is zero, that means that z-scores are measuring deviation from the mean. If a z-score is below the mean, it means that the corresponding raw score is below the mean, which results in a negative z-score.
Z-scores that fall below the mean are negative. Z-scores are statistical values that allow analysts to transform raw data into standardized units that make comparisons possible among scores in the distribution. The formula for calculating the z-score is: z = (X-μ)/σThe z-score can be either negative, zero, or positive, depending on whether the value is below, equal to, or above the mean, respectively. Because the mean of all z-scores is always zero, z-scores can be used to measure deviation from the mean, with a negative z-score indicating that the corresponding raw score is below the mean. Z-scores allow us to compare the scores in the distribution by converting them into standard units. Negative z-scores are associated with scores that are below the mean. Conversely, positive z-scores are associated with scores that are above the mean.
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If 8 people consisting of 4 couples are randomly arranged in a row, find the probability that no person is next to their partner.
Answers
The Probability that no person is next to their partner is 104/105.
Probability of an event E represented by P(E) can be defined as (The number of favorable outcomes )/(Total number of outcomes).
Permutations is defined as arrangement of elements/objects in a particular way.
According to the question ,
8 people can be arranged in 8! ways = 40320ways
First let us find the probability that person is next to their partner.
4 couples can be arranged in 4! ways
and the couple itself can be arranged in 2! ways
Since there are 4 couples ,
Number of arrangement = \(4!(2!)^{4}\)
=24x16
=384 ways
Probability that person is next to their partner = \(\frac{384}{40320} =\frac{1}{105}\)
and the probability that no person is next to their partner is \(1-\frac{1}{105} =\frac{104}{105}\).
Therefore , the probability that no person is next to their partner is \(\frac{104}{105}\).
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AGAIN HELP!!!!!!!!!!!!!!!STILL TRYING TO TEACH MYSELF LOL
Answers
= 70
HELPPP ASAP PLZ!!!! Will mark you brainliest
Answers
Answer: B and D
Step-by-step explanation:
A. States that the length of AB is -3. No, it should be 3. So that is incorrect.
B. States that the absolute value of 6-1 is the length of BC. It is 5. That is correct.
C. AB + AC = 3+8 = 11. BC is 5. That is incorrect.
D. AB + BC = AC. 3+5= 8. AC is 8. That is correct.
E. The length is not 6. It is 5. That is incorrect.
Hope that helps!
Meredith rides her bicycle for 8 hours and is 74 miles from her house. After riding for 11 hours, she is 101 miles away. What is Meredith's average rate during her trip
Answers
Meredith's average rate during her trip is 33.67 miles/hour.
Average rate or average speed is a measure of the overall rate of change of a quantity over a given time period. It represents the average amount of a quantity that changes per unit of time. It is calculated by dividing the total change in the quantity by the total time elapsed.
Mathematically, the formula for average rate is:
Average Rate = (Change in Quantity) / (Change in Time)
This formula applies to various scenarios, such as distances traveled, amounts produced, or any other measurable quantity. It is important to note that the average rate is based on the assumption of a constant rate of change throughout the entire time interval.
Average rate = Total distance / Total time
= 101 miles / 3 hours
≈ 33.67 miles/hour
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FILL IN THE BLANK a/an ____________________________ diagram can be used to show how the tables in a database are defined and related..
Answers
A/an database schema diagram can be used to show how the tables in a database are defined and related.
This type of diagram provides a visual representation of the structure and organization of the database. It illustrates the tables ,and their attributes, and the relationships between them.
The database schema diagram helps in the understanding the logical design of the database, including primary keys, for foreign keys, and the connections between different tables. It allows developers, database administrators, and stakeholders to visualize the database structure and serves it as a reference for in designing, modifying, and querying the database effectively.
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a sample consists of the following data: 7, 11, 12, 18, 20, 22, 43. Using the three standard deviation criterion, the last observation (x=43) would be considered an outlier
a. true
b. false
Answers
The statement "Using the three standard deviation criterion, the last observation (x=43) would be considered an outlier" is true.
Given data:
7, 11, 12, 18, 20, 22, 43.
To find out whether the last observation is an outlier or not, let's use the three standard deviation criterion.
That is, if a data value is more than three standard deviations from the mean, then it is considered an outlier.
The formula to find standard deviation is:
S.D = \sqrt{\frac{\sum_{i=1}^{N}(x_i-\bar{x})^2}{N-1}}
Where, N = sample size,
x = each value of the data set,
\bar{x} = mean of the data set
To find the mean of the given data set, add all the numbers and divide the sum by the number of terms:
Mean = $\frac{7+11+12+18+20+22+43}{7}$
= $\frac{133}{7}$
= 19
Now, calculate the standard deviation:
$(7-19)^2 + (11-19)^2 + (12-19)^2 + (18-19)^2 + (20-19)^2 + (22-19)^2 + (43-19)^2$= 1442S.D
= $\sqrt{\frac{1442}{7-1}}$
≈ 10.31
To determine whether the value of x = 43 is an outlier, we need to compare it with the mean and the standard deviation.
Therefore, compute the z-score for the last observation (x=43).Z-score = $\frac{x-\bar{x}}{S.D}$
= $\frac{43-19}{10.31}$
= 2.32
Since the absolute value of z-score > 3, the value of x = 43 is considered an outlier.
Therefore, the statement "Using the three standard deviation criterion, the last observation (x=43) would be considered an outlier" is true.
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Aubree invests money in an account paying simple interest. She invests $30 and no
money is added or removed from the investment. After one year, she has $32.40.
What is the simple percent interest per year?
Answers
Answer:
8%
Step-by-step explanation:
$30 divided by 100= 0.3/1%
0.3x8=$2.40
$30+$2.4=32.40
The quality control manager at a computer manufacturing company believes that the mean life of a computer is 80 months, with a variance of 6464. If he is correct, what is the probability that the mean of a sample of 7777 computers would be less than 82.5982.59 months
Answers
The probability that the mean of a sample of 7777 computers would be less than 82.59 months is 100%, or close to 1.
The probability that the mean of a sample of 7777 computers would be less than 82.5982.59 months, assuming a mean life of 80 months and a variance of 6464, can be calculated using the central limit theorem and the standard normal distribution.
First, we calculate the standard error of the mean using the formula:
standard error of the mean = σ/√n
where σ is the population standard deviation, n is the sample size.
Here, σ² = 6464, so σ = √6464 = 80.3
n = 7777
standard error of the mean = 80.3/√7777 ≈ 0.907
Next, we calculate the z-score using the formula:
z = (\(\bar{x}\) - μ) / (σ/√n)
where \(\bar{x}\) is the sample mean, μ is the population mean, σ is the population standard deviation, n is the sample size.
Here, \(\bar{x}\) = 82.59, μ = 80, σ = 80.3, n = 7777
z = (82.59 - 80) / (80.3/√7777) ≈ 8.6
We find that the probability of z being less than 8.6 is very close to 1, or 100%.
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v-7/3 = 0
..........
Answers
Answer:
v = 7/3
Step-by-step explanation:
Add 7/3 to both sides of the equation:
v -7/3 +7/3 = +7/3
v = 7/3
_____
If you intend ...
\(\dfrac{v-7}{3}=0\)
then parentheses are needed:
(v-7)/3 = 0
The value of v that makes this true is v=7.
Which expression represents the surface area of the
pyramid?
A 4+6+6+6+6
B 3x2
C 4+3+3+3+3
D 4x3
Answers
Answer:
the answer is a I think your welcome
Surface area is the sum of the area of faces.
This means that:
Surface area of prism = 4(area of triangle) + area of squareFinding the surface area of the prism.
=> Surface area of square based prism = 4(1/2 x 2 x 3) + 2 x 2=> Surface area of square based prism = 4(3) + 4=> Surface area of square based prism = 12 + 4 = 16 units²Verifying Option A
4 + 6 + 6 + 6 + 6=> 4 + 24=> 28 units² ≠ 16 units²Thus, Option A is incorrect.
Verifying Option B
3 x 2=> 6 units² ≠ 16 units²Thus, Option B is incorrect.
Verifying Option C
4 + 3 + 3 + 3 + 3=> 4 + 12=> 16 units² = 16 units²Thus, Option C is correct.
Verifying Option D
4 x 3=> 12 units² ≠ 16 units²Thus, Option D is incorrect.
In conclusion...
Option C is correct.
Three equal circles with radius r are drawn as shown, each with its centre on the circumference of the other two circles. A, B and C are the centres of the three circles. Prove that an expression for the area of the shaded region is:
A = r^2 / 2 (Pi - _/3)
Answers
The shaded region is composed of three separate circles with their centers located at A, B, and C. By the definition of a circle, the circumference of each circle is 2πr, where r is the radius. The area of a circle can be calculated by A = πr^2.
Therefore, the area of the shaded region can be calculated using the following equation:
A = (area of circle A) + (area of circle B) + (area of circle C)
A = πr^2 + πr^2 + πr^2
A = 3πr^2
However, the shaded region is also composed of three 120° central angles which have intercepted arcs of the three circles. The length of each arc is (2πr)/3. Therefore, the area of the shaded region can be calculated using the following equation:
A = (area of circle A) - (area of intercepted arcs)
A = 3πr^2 - (2πr/3)^2
A = 3πr^2 - (4πr^2/9)
A = r^2 / 2 (π - _/3)
Therefore, the expression for the area of the shaded region is A = r^2 / 2 (π - _/3).
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2 3 6 4 5 4 4 4 3 2 5 3 6 6 4 1 1 2 3 2 4 2 1 4 2
What is the mean of this data set?
Answers
I need help it’s due today please help
Answers
Answer:
$1050
Step-by-step explanation:
Simple interest:
I = prt
I = 5000 * 0.07 * 3
I = 1050
8 plus the quotient of a number and 3 is negative 2
Answers
Answer:
Step-by-step explanation: x = the number
8 + x/3 = -2
Multiply both sides by 3 and get
24 + x = -6
Solve for x:
x = -30
PLEASE HELP!!
I'll mark brainliest! Please put it neatly so I can see :)
Answers
Answer: you can put it into addition form or multiplication just take the number from that place as the right amount of zeros behind it and you should be good so on 6 it could be 90 or if your suppose to be putting it in multiplication it would be 9x10
Step-by-step explanation:
Answer:
read below
Step-by-step explanation:
decomposed answ:
6. 9*10
7. 4*10
8. 89*10
9. 3*100
10. 7*1000
11. 37*100
decomposition:
12. 56*10
13. 43*100
14. 60*100