We have proved that a^1 ≡ a^(mod n) and b ≡ c (mod ordₙ(a)).
To prove the given statements, we will use the properties of congruence and the concept of the order of an element modulo n.
Statement 1: a^1 ≡ a^(mod n)
Let's consider a positive integer k such that k ≡ 1 (mod φ(n)), where φ(n) represents Euler's totient function. By Euler's theorem, we know that a^φ(n) ≡ 1 (mod n). Therefore, we can rewrite k as k = 1 + mφ(n), where m is an integer. Now, we can raise both sides of the congruence to the power of a, yielding a^k ≡ a^(1+mφ(n)) (mod n). By applying the properties of congruence, we have a^k ≡ a^1 ⋅ (a^φ(n))^m ≡ a (mod n). Hence, a^1 ≡ a^(mod n).
Statement 2: b ≡ c (mod ordₙ(a))
Let ordₙ(a) denote the order of a modulo n. By definition, ordₙ(a) is the smallest positive integer k such that a^k ≡ 1 (mod n). Since b ≡ c (mod ordₙ(a)), we can express b as b = c + k⋅ordₙ(a), where k is an integer. Then, we have a^b ≡ a^(c+k⋅ordₙ(a)) ≡ a^c ⋅ (a^(ordₙ(a)))^k ≡ a^c ⋅ 1^k ≡ a^c (mod n), which implies b ≡ c (mod ordₙ(a)).
In conclusion, we have proved that a^1 ≡ a^(mod n) and b ≡ c (mod ordₙ(a)).
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The water used by the 12 students during the experiment was poured from a beaker after the water was poured 1/4 gallon of water was left in the beaker What was the total number of fluid ounces of water in the beaker before the water was poured by the 12 students
Answer:
128 fluid ounces
Step-by-step explanation:
We were told that:
During the experiment was poured from a beaker after the water was poured 1/4 gallon of water was left in the beaker
Hence, this means that the total gallon of water in the beaker is 1 gallon
Convert 1 gallon to fluid ounces
1 gallon =128 fluid ounces
Therefore, the total number of fluid ounces of water in the beaker before the water was poured by the 12 students is 128 fluid ounces.
Find the general solution of the following second order differential equation. y" + 6y' +9y = e^-3x Inx.
General solution of the second order differential equation:
-6Ae^(-3x) / x + 9Bxe^(-3x) * ln(x) - Be^(-3x) / x^2
To find the general solution of the given second-order differential equation: y" + 6y' + 9y = e^(-3x) * ln(x)
We will use the method of undetermined coefficients to find a particular solution and then combine it with the complementary solution to obtain the general solution.
Step 1: Finding the particular solution
Since e^(-3x) * ln(x) is a product of exponential and logarithmic functions, we assume a particular solution in the form of:
yp = (A + Bx) * e^(-3x) * ln(x)
where A and B are constants to be determined.
Step 2: Find the first and second derivatives of yp.
yp' = (Ae^(-3x) * ln(x) - 3(A + Bx)e^(-3x) * ln(x) + (A + Bx) * e^(-3x) / x
yp" = (Ae^(-3x) * ln(x) - 3(A + Bx)e^(-3x) * ln(x) + (A + Bx) * e^(-3x) / x)'
yp" = (-9(A + Bx)e^(-3x) * ln(x) + 3(A + Bx)e^(-3x) * ln(x) - 3(A + Bx)e^(-3x) / x + (A + Bx) * (-e^(-3x) / x^2 + e^(-3x) / x))
Simplifying, we have:
yp" = -9(A + Bx)e^(-3x) * ln(x) - 6(A + Bx)e^(-3x) / x + (A + Bx) * (-e^(-3x) / x^2 + e^(-3x) / x)
Step 3: Substitute yp, yp', and yp" into the original differential equation to find the values of A and B.
yp" + 6yp' + 9yp = e^(-3x) * ln(x)
Substituting the expressions we found for yp and its derivatives:
(-9(A + Bx)e^(-3x) * ln(x) - 6(A + Bx)e^(-3x) / x + (A + Bx) * (-e^(-3x) / x^2 + e^(-3x) / x))
6((Ae^(-3x) * ln(x) - 3(A + Bx)e^(-3x) * ln(x) + (A + Bx) * e^(-3x) / x))
9((A + Bx) * e^(-3x) * ln(x))
= e^(-3x) * ln(x)
Expanding and simplifying, we get:
-9Ae^(-3x) * ln(x) + 3Bxe^(-3x) * ln(x) - 6Ae^(-3x) / x + 6Bxe^(-3x) / x + Ae^(-3x) / x - Be^(-3x) / x^2 + Be^(-3x) / x + 9Ae^(-3x) * ln(x) + 9Bxe^(-3x) * ln(x)
= e^(-3x) * ln(x)
Combining like terms, we have:
-6Ae^(-3x) / x + 9Bxe^(-3x) * ln(x) - Be^(-3x) / x^2
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2.46 strong association but no correlation. here is a data set that illustrates an important point about correlation: corr x 25 35 45 55 65 y 10 30 50 30 10 (a) make a scatterplot of y versus x. (b) describe the relationship between y and x. is it weak or strong? is it linear? (c) find the correlation between y and x. (d) what important point about correlation does this exercise illustrate?
a. In the picture we can see that the scatterplot is given for the data in the given table.
b. It is not linear as we can see from scatterplot.
c. The correlation between y and x is 0.
d. when r = 0 there is no relationship between x and y.
Given that,
The data is given in the table.
We know that,
a. We have to make a scatterplot of y versus x.
In the picture we can see that the scatterplot is given for the data in the given table.
b. We have to describe the relationship between y and x.
When x increases y increases upto certain point after that y start to decrease but x is increases only
Therefore, it is not linear as we can see from scatterplot.
c. We have to find the correlation between y and x.
The formula for the correlation coefficient is
r = [tex]\frac{n\times \sum XY-\sum X \times\sum Y}{\sqrt{[n\sum X^2 - (\sum X)^2][n\sum Y^2 - (\sum Y)^2]} }[/tex]
Now we find summation of all X, Y and XY, [tex]X^2[/tex] and [tex]Y^2[/tex]
X Y XY [tex]X^2[/tex] [tex]Y^2[/tex]
25 10 250 625 100
35 30 1050 1225 900
45 50 2250 2025 2500
55 30 1650 3025 900
65 10 650 4225 100
Now, ∑X = 225
∑Y = 130
∑XY = 5850
∑[tex]X^2[/tex] = 11125
∑[tex]Y^2[/tex] = 4500
Now, Substitute the values in the formula
r = [tex]\frac{5\times 5850-225 \times130}{\sqrt{[5\times 11125 - (225)^2][5\times 4500 - (130)^2]} }[/tex]
r = 0
Therefore, The correlation between y and x is 0.
d. We have to find what important point about correlation does this exercise illustrate.
Therefore, when r = 0 there is no relationship between x and y.
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PLEASE HELP ILL GIVE BRAINLIEST
WHATS THE PERCENT CHANGE SHOW WORK
Answer:
34.55556%
Step-by-step explanation:
Identify the values of A, B, and C
5x2 + x - 18 = - 9x + 3x2
A=2, B=10, C=-18
A=2, B=9, C=-18
A=5, B=1, C=-18
A=8, B=-8, C=-18
Answer:
A=2, B=10, C=-18
The answer is the first one.
Step-by-step explanation:
[tex]5x^{2} -3x^{2} +x+9x-18=0[/tex]
[tex]2x^{2} +10x-18=0[/tex]
Formula: [tex]f(x)=ax^{2} +bx+c[/tex]
Identify the location of the point (6, -2). A. P B. Q C. R D. S
Answer:
Step-by-step explanation:
Which of the following represents y < 2x - 3?
help a girl out please
Answer:
A
Step-by-step explanation:
the y intercept (where x is 0) should be -3 because of y<3x-3
Solve the differential equation (D^2 + 4)y=6 sin2x +3x^2 =
The solution to the differential equation (D^2 + 4)y = 6 sin(2x) + 3x^2 is y = (3/4)x^2 + A sin(2x) + B cos(2x).
To solve the given differential equation (D^2 + 4)y = 6 sin(2x) + 3x^2, where D represents the derivative operator, we can use the method of undetermined coefficients.
First, we find the general solution to the homogeneous equation (D^2 + 4)y = 0. The characteristic equation is r^2 + 4 = 0, which has complex roots ±2i. Therefore, the general solution to the homogeneous equation is y_h = A sin(2x) + B cos(2x), where A and B are constants.Next, we find a particular solution to the non-homogeneous equation. By inspection, we can guess that y_p = (3/4)x^2 is a particular solution.Finally, the general solution to the non-homogeneous equation is the sum of the homogeneous and particular solutions:y = y_h + y_p
y = A sin(2x) + B cos(2x) + (3/4)x^2
Here, A and B are arbitrary constants that can be determined by applying initial or boundary conditions, if given.
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This is confusing can you help? NO LINKS!!!
Answer:
huh?
Step-by-step explanation:
did you forget a pic?
Answer:
whatdo u need help with love?
Step-by-step explanation:
Evaluate the expression when b=4 and x=-2 .
b-4x
Answer:
12
Step-by-step explanation:
b-4x
Plug in 4 as b and -2 as x
= (4)-4(-2)
Multiply 4 and 2
= 4-(-8)
Two negatives make a positive
= 4+8
= 12
I hope this helps!
Which set of numbers is infinite?
A) integers between -4 and 7
B) whole numbers between 1 and 17
C) natural numbers between 5 and 10
D) irrational numbers between 10 and 30
Answer:
D). irrational numbers between 10 and 30
Does anyone know how to work this out step by step?
Answer:
[tex]2x^{3} +10[/tex]
Step-by-step explanation:
[tex]x*2x*x+10=2x^{3} +10[/tex]
a manufacturer knows that their items have a normally distributed lifespan, with a mean of 4.4 years, and standard deviation of 1.2 years.the 7% of items with the shortest lifespan will last less than how many years?
Using the standard deviation, mean, and z-score, the 7% of items with the shortest lifespan will last less than approximately 1.59 years.
What is 7th percentile of items with the shortest lifespan?To find the number of years that the 7% of items with the shortest lifespan will last, we need to determine the z-score corresponding to the 7th percentile of the normal distribution.
Step 1: Convert the given percentile to a z-score using the standard normal distribution table or a statistical calculator. The 7th percentile corresponds to a z-score of approximately -1.405.
Step 2: Use the formula for z-score to find the corresponding value in terms of years:
x = μ + z * σ
where x is the value we are looking for, μ is the mean, z is the z-score, and σ is the standard deviation.
Plugging in the values:
x = 4.4 + (-1.405) * 1.2
x = 1.59 years
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What is the slope and y-intercept of 9x - 3y = 15?
NO FILES PEOPLE!
slope is 3
y-intercept is 0,-5
If sin A=0.7986, then the measure of Đ A, to the nearest degree is?
Answer:
B
Step-by-step explanation:
A will equal the sine inverse (arc sine or sin^-1 ) of 0.3571. Using a calculator, sin^-1(0.3751) = 22.03 degrees (B)
approximate the area under the curve graphed below from x = 2 x=2 to x = 5 x=5 using a left endpoint approximation with 3 subdivisions.
The approximate area under the curve from x = 2 to x = 5, using a left endpoint approximation with 3 subdivisions, is 13.5 square units.
To approximate the area under the curve, we divide the interval from x = 2 to x = 5 into three equal subdivisions, each with a width of (5 - 2) / 3 = 1. The left endpoint approximation involves using the leftmost point of each subdivision to approximate the height of the curve.
In this case, we evaluate the function at x = 2, x = 3, and x = 4, and use these values as the heights of the rectangles. The width of each rectangle is 1, so the areas of the rectangles are calculated as follows:
Rectangle 1: Height = f(2) = 2, Area = 1 * 2 = 2 square units.
Rectangle 2: Height = f(3) = 4, Area = 1 * 4 = 4 square units.
Rectangle 3: Height = f(4) = 7, Area = 1 * 7 = 7 square units.
Finally, we add up the areas of the three rectangles to obtain the approximate area under the curve: 2 + 4 + 7 = 13 square units. Therefore, the approximate area under the curve from x = 2 to x = 5 using a left endpoint approximation with 3 subdivisions is 13.5 square units.
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Peter is making an "X marks the spot" flag for a treasure hunt. The flag is made of a square white flag with sides of 12 centimeters. He will make the "X" by stretching red ribbon diagonally from corner to corner.
Answer:
33.94 cms of ribbon
Step-by-step explanation:
Because you need two diagonals to form the x, therefore the amount of ribbon needed is the sum of the distance of both diagonals.
When crossing the diagonal, a rectangular angle is formed, where the diagonal would be the hypotenuse, we know that the distance of the hypotenuse can be calculated by means of the legs, which we know its value (12):
d ^ 2 = a ^ 2 + b ^ 2
a = b = 12
d ^ 2 = 12 ^ 2 + 12 ^ 2
d ^ 2 = 288
d = 288 ^ (1/2)
d = 16.97
16.97 cm is what it measures, a diagonal, therefore tape is needed:
16.97 * 2 = 33.94
A total of 33.94 cms of ribbon is needed
Answer from jmonterrozar
Regan cycles 78 miles in 6 hours. His average speed for the first 30 miles is 15 miles per hour. Work out Regan's average speed for the last 48 miles.
Answer:
12 mph
Step-by-step explanation:
Given that:
Total distance traveled = 78 miles
Time taken, t = 6 hours
Average speed for first 30 miles = 15 mph
Time taken = distance / speed
Time taken = 30 / 15
Time taken = 2 hours
Average speed for last 48 miles = x
Time taken to travel last 48 miles = (6 - 2) = 4 hours
Average speed for last 48 miles :
Distance traveled / time taken
48 miles / 4 hours
= 12 mph
What should be done to both sides of the equation in order to solve y - 4.5 = 12.2?
To solve the equation y - 4.5 = 12.2 we have to add 4.5 on both sides of the equation.
What is Equation?Two or more expressions with an Equal sign is called as Equation.
The given equation is y - 4.5 = 12.2
y minus four point five equal to twelve point two.
In the equation y is the variable and minus is the operator in the equation.
To solve the equation we have to isolate the variable y.
To isolate the variable y we have to add 4.5 on both sides of the equation
y-4.5+4.5=12.2+4.5
y=16.7
Hence, to solve the equation y - 4.5 = 12.2 we have to add 4.5 on both sides of the equation.
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Does the equation 3(2x−1)+5=6(x+1) have one, none, or an infinite amount of solutions?
Answer: No solutions
Step-by-step explanation: If you solve the problem all the way, you get 0 = 4 which is not valid so there is simply no solution
The given equation 3(2x−1)+5=6(x+1) has no solution. Option B is correct.
What is simplification?The process in mathematics to operate and interpret the function to make the function or expression simple or more understandable is called simplifying and the process is called simplification.
Here,
Given equation ⇒
⇒= 3(2x−1)+5=6(x+1)
Simplify the above expression,
⇒ 6x - 3 + 5 = 6x + 1
⇒ 2 ≠ 1
Thus, the given equation has no solution.
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Question 3 of 10
Which of the following are exterior angles? Check all that apply.
6
DA. 26
B. 24
C. 23
OD. 22
OE. Z1
O F. 25
Answer:
I believe the answer to be 5 and 4
The exterior angles of the given triangle are angles 4 and 5
What is triangle?A triangle is a polygon with three sides, angles and vertices.
Given that, a triangle, with angles, 1, 2, 3, 4, 5 we need to find the exterior angles,
An exterior angle of a polygon is the angle that lies outsides of the polygon,
Here, we can see, that only two angles 4 and 5 lies outsides of the triangles
Therefore, the exterior angles are 4 and 5
Hence, the exterior angles of the given triangle are angles 4 and 5
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PLEASE HELP ME!!!! In each diagram AB is tangent to C at B. Find the value of x.
Simplify. simplify simplify
[tex]\huge\text{Hey there!}[/tex]
[tex]\mathsf{(\dfrac{4}{9})^2}[/tex]
[tex]\mathsf{= \dfrac{4}{9}\times\dfrac{4}{9}}[/tex]
[tex]\mathsf{= \dfrac{4^2}{9^2}}[/tex]
[tex]\mathsf{\dfrac{4\times4}{9\times9}}[/tex]
[tex]\mathsf{4 \times 4 = \bold{16}\leftarrow \underline{Numerator}}[/tex]
[tex]\mathsf{9\times9 = \bf 81}\leftarrow\mathsf{\underline{Denominator}}[/tex]
[tex]\boxed{\boxed{\large\textsf{Answer: \boxed{{\bf \dfrac{16}{81}}}}}}\huge\checkmark[/tex]
[tex]\text{Good luck on your assignment and enjoy your day!}[/tex]
~[tex]\frak{Amphitrite1040:)}[/tex]
A rectangular prism has a volume of 900 cubic units. The prism has a length of 25 units and a width of 12 units. Which equation could be used to find h, the height of the prism?
Answer:
52 cubic units
Step-by-step explanation:
got it right on edg
Answer:
37h 900
Step-by-step explanation:
1 point
Finish the similarity statement. (Note: your answer will be the 3 letters of
the other triangle. You must place them in the correct order and use
CAPITAL letters!) APQR-4
R
4 in
6 in
VA
8 in
4 in
3 in
2 in
YOUR ANSWER IS ANGLE BAC
Find the product.
-7(-a 2)(-5)
Answer:
3
5
(
−
2
)
Step-by-step explanation:
Simplify then multiple the numbers
Given that coule.us) - EILE DE M2)]. lajure the linearity rule and & (c) = c. to derive the equation for constate) in ternis of EA), Mj and H2(erive expression for cours, 34%, and 22 are independent random vartolus. f(x)= [ (x for 2 exc4 = 56) o elsewhere for a continuon ona random variable &. (a) Compute. P/2 ex <3). (6) Compute Elx), the mean of t. (8) Given (6) For some other random variable & My (t) = e. Determine the mean Ele) for this other random variable. (5* +32+) P.
(a) The probability that X is less than 3, P(X < 3), is 0.
(b) The mean of X, denoted as E(X), is 71/24, which is approximately 2.9583 when rounded off to four decimal places.
(c) Given Y = e^X, the mean of Y, denoted as E(Y), is approximately 15.75 when rounded off to two decimal places.
(a) It is required to compute P(X<3). Since the range for which f(x) is not equal to 0, is the interval from 2 to 4 for f(x), the probability that X is less than 3 is 0.
Similarly, for X > 4, P(X > 4) = 0.
P(2 ≤ X ≤ 4) = ∫f(x)dx from 2 to 4= ∫ (x/24 - 7/3) dx from 2 to 4= [x^2/(2 × 24) - 7x/3] from 2 to 4= 1/4 - 28/3 + 8/(2 × 24) + 14/3= 71/24= 2.9583(rounded off to four decimal places)
(b) The mean of X can be computed as follows:
E(X) = ∫xf(x)dx from -∞ to ∞= ∫ (x/24 - 7/3) dx from 2 to 4= [x^2/(2 × 24) - 7x/3] from 2 to 4= 1/4 - 28/3 + 8/(2 × 24) + 14/3= 71/24= 2.9583(rounded off to four decimal places)
(c) Y = e^X
The mean of Y can be computed as follows:
E(Y) = E(e^X)= ∫ e^x f(x) dx from -∞ to ∞= ∫ e^x (x/24 - 7/3) dx from 2 to 4= [e^x (x - 31)/(24)] from 2 to 4= (e^4/6 - 31e^4/24 - e^2/6 + 31e^2/24) ≈ 15.75(rounded off to two decimal places).
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sara bought x pounds of chocolate covered raisins, which sell for $1.50 a pound, and y pounds of yogurt covered raisins, which sell for $1.20 a pound. sara bought a total of 40 pounds of the two types of raisins for a total of $51.90." help me write a system of equations to model this scenario
HELP ASAP, due today.
Answer:
(A) B h(x)= -2x-5.5
(B) The y intercept is (0,-5.5)
(C) The rate of change is -2
(D) The x intercept is (-2.75,0)
Solve for x. Please help me I am confused.
Answer:
72
Step-by-step explanation:
Multiply all the numbers, and you get the answer