The expected value of the square of the distance of the knight after 2016 moves is approximately 462.05.
Expected Square Distance of KnightThe expected value of the square of the distance of the knight from the origin after n moves can be calculated using the following formula:
E(d²) = (1/8) * (2 * (E(d²) - 1) + 5)
where d is the distance from the origin, and E(d²) is the expected value of d².
Using this formula, the expected value of the square of the distance after 2016 moves can be calculated iteratively.
Starting with an initial estimate of E(d²), we can use the formula to calculate a better estimate in each iteration. This process can be repeated until the estimate converges to a value within a desired accuracy.
Using this approach, the expected value of the square of the distance after 2016 moves can be approximated to be 462.05.
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increase 200 by 50%
pls give simple worrking out :)
Answer:
300
Step-by-step explanation:
1.5*200 = 300
Cool.
The percentage of battery remaining, y, on a tablet's battery after x hours can be represented by the given graph.
What is the meaning of the y-intercept in the context of the problem?
The tablet will not have any battery remaining after 8 hours.
The tablet loses 8 percent of battery every hour.
After 80 hours, the tablet will have 0 percent of battery left.
The tablet starts with 80 percent of battery remaining.
The meaning of the y-intercept of the linear function is given as follows:
The tablet starts with 80 percent of battery remaining.
How to interpret the definition of the linear function?The slope-intercept definition of a linear function is given as follows:
y = mx + b.
In which:
The slope m represents the rate of change of the output variable relative to the input variable.The intercept b represents the value of y when the graph of the function touches of crosses the y-axis.The graph of the function touches the y-axis at y = 80, hence the intercept b is given as follows:
b = 80.
As the output variable is the battery percentage, it represents the initial battery percentage.
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The meaning of the y-intercept of the linear function is given as follows:
The tablet starts with 80 percent of battery remaining.
The slope-intercept definition of a linear function is given as follows:
y = mx + b.
In which:
The slope m represents the rate of change of the output variable relative to the input variable.
The intercept b represents the value of y when the graph of the function touches of crosses the y-axis.
The graph of the function touches the y-axis at y = 80, hence the intercept b is given as follows:
b = 80.
As the output variable is the battery percentage, it represents the initial battery percentage.
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Solve 1 and 2 please help
Step-by-step explanation:
g(x) is f(x) shifted up by 2 units.
that is simply expressed as
g(x) = f(x) + 2
h(x) if f(x) shifted left by 1 unit.
that is trickier.
h(x) = f(x + 1)
why ?
every result that f(x) had at x+1, is now happening for h(x) at x ("earlier", shifted to the left). so, the function result for f(x+1) is the function result for h(x).
and therefore,
h(x) = f(x + 1)
FYI
a shift to the right by 1 unit would be
i(x) = f(x - 1)
for the same reasons as for the shift to the left, just in the other direction.
A scooter costs a retailer $75, but it sells for $125 at the local store. What is the percent markup? Round to the nearest whole percent.
Total percent markup in the price of the scooter is 66.67 % ≈ 67%
What is Percentage ?The Latin phrase "per centum," which meaning "by the hundred," was the source of the English word "percentage." Fractions having a denominator of 100 are called percentages. In other words, it is a relationship where the value of the entire is always assumed to be 100.
A percentage is a ratio or fraction where the full value is always 100. Sam, for instance, would have received 30 out of a possible 100 points if he had received 30% on his arithmetic test. In ratio form, it is expressed as 30:100 and in fraction form as 30/100.
Finding the percentage of a whole in terms of 100 is what percentage calculation refers to. A percentage can be found in one of two ways:
use the unitary approach.
by adjusting the fraction's denominator to 100.
It should be noted that when the denominator is not a factor of 100, the second technique of percentage calculation is not applied. In these situations, we employ the unitary technique.
Cost of Scooter at retailer is $ 75
Cost of Scooter at local Store is $125
Total price markup = 125- 75 = $50
Total percent markup is = 50/75 * 100 = 66.67 % ≈ 67%(approx.)
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27. A company wants to buy phones for its employees. Employee executives will get smart
phones at a cost of $250 each and regular employees will get flip phones at a cost of $50
each. If the company spends $4,000 on a total 40 phones, and every employee gets a
phone, how many more regular employees than executives work at the company?
A) 20
B) 25
C) 30
D) 35
The number of regular employees that got a phone is 39
The number of executive employees that got a phone is 1
How to calculate the number of regular and executive employees that got phones?Let x represent the number of regular employees
Let y represent the number of executive employees
x + y= 40.......equation 1
50x + 250y= 4000...........equation 2
From equation 1
x= 40-y
Substitute 40-y for x in equation 2
50(40-y) + 250y= 4000
2000- 50y + 250y= 4000
2000 + 2000y= 4000
2000y= 4000-2000
2000y= 2000
y = 1
Substitute 1 for y in equation 1
x + y= 40
x + 1= 40
x= 40-1
x= 39
Hence there are 39 regular employees that got a phone and 1 executive employee that got a phone.
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how many equilateral triangles in the plane have at least two of the points in the triangular lattice below as vertices?
In the triangular lattice below, there are an infinite number of equilateral triangles that have at least two of the points as vertices.
This is because the triangular lattice is made up of an infinite number of equilateral triangles of various sizes and orientations. Any two points in the lattice form the vertices of an equilateral triangle and any other point in the lattice can be used as the third vertex.
An equilateral triangle is a triangle in which all three sides are equal in length. It is also equiangular, meaning that all three angles are equal in measure. In the plane, all three vertices of an equilateral triangle can be connected to form a single closed figure. Equilateral triangles can also be used to form periodic patterns, such as the triangular lattice shown in the question.
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describe the line in coordinate form passing through the point (−4,−6,6) in the direction of . (write your solution using the form (*,*,*). use symbolic notation and fractions where needed.)
The line in coordinate form passing through the point (−4,−6,6) in the direction is x - y - z + 4 = 0
The term called coordinates in math is called a system that uses one or more numbers, or coordinates and it is uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space.
Here we have the line in coordinate form passing through the point (−4,−6,6) in the direction.
As we all know that the equation of line through the point (-4, -6, 6) is written as,
=> ( x - (-4)) = (y - (-6)) = (z - 6)
When we simplify this one then we get,
=> x + 4 = y + 6 = x - 6
=> x - y - z + 4 - 6 + 6
=> x - y - z + 4 = 0
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PLS HELP ASAP Solve 0.408 ÷ 6.
0.068
0.680
2.448
−5.592
Answer: 0.068
Step-by-step explanation:
i used a calculator so I think it's right
I really need help with this question please and thank you .
how to find where derivative is positive, negative, and undeffined
The sign of the derivative of a function at a certain point indicates the nature of the function's change at that point.
If the derivative is positive at a point, the function is increasing at that point. If the derivative is negative at a point, the function is decreasing at that point. If the derivative is undefined at a point, there may be a sharp turn or a discontinuity at that point.
Here's a more specific method to determine the sign of the derivative at a point:
Find the first derivative of the function.Pick a small interval around the point in question and divide it into smaller sub-intervals.Evaluate the first derivative at the endpoints of each sub-interval.If the derivative is positive at one endpoint and negative at the other, the derivative is zero at some point in the interval, which indicates a maximum or minimum.If the derivative is positive on both endpoints, it is positive on the whole interval and therefore the function is increasing.If the derivative is negative on both endpoints, it is negative on the whole interval and therefore the function is decreasing.Repeat steps 3-6 for smaller sub-intervals to get a more accurate result.It's also possible to determine the sign of the derivative by analyzing the graph of the function, or by using calculus theorems such as the First Derivative Test or the Second Derivative Test.
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F(x) = x^3 2x^2 - 6x 9
When f(x) i divided by x a, the remainder i R. When f(x) i divided by x - a, the remainder i 2R. A) Show that 3a^3 - 2a^2 - 18a - 9 = 0. B) Solve the equation in part a completely
B) The Rational Root Theorem may be used to identify the potential rational roots of the equation in Part A.
The divisors of 9 with the opposite sign are possibly rational roots.
Then, for each rational root, we can see if it is a factor of the polynomial.
A) The remaining of f(x) divided by (x - a) is R, while the remainder of f(x) divided by (x - a) is 2R.
As a result, we may write the following polynomials:
f(x) = (x - a) (x - a)
Q(x) + R f(x) = (x - a)
Q(x) + 2R
Where Q(x) represents the quotient and R represents the remainder.
By combining the two polynomials, we obtain:
(x - a)
Q(x) + R = (x - a) (x - a)
Q(x) + 2R
When we subtract (x - a)Q(x) from both sides, we get:
R = 2R
R = 0 when we solve for R
As a result, the remaining when f(x) is divided by (x - a) is zero.
That is, (x - a) is a factor of f(x), and f(x) is represented as:
f(x) = (x - a)(x^2 + bx + c)
Comparing the coefficients, we get:
a = -a
b = 2a
c = -9
So we have the following equation:
x^3 - 2x^2 - 18x - 9 = 0
B) If it is a factor, we may use synthetic division to divide the polynomial by the factor, resulting in a new polynomial, which we can continue until we have a polynomial with no rational roots. We can utilize numerical methods such as the bisection method or the Newton-Raphson method to thoroughly solve the problem.
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In the rectangle below, FH-7x-2, EI-20, and mZIFG=39°,
Find the value of x and mZEIH.
H
F
G
Rectangle EFGH is assumed, the value of x and m<EIH is 102 degree
What is the value of x and y in the given figure if EFGH is rectangle?Rectangle EFGH is assumed. Each side of a rectangle has an equal length. Each side of a rectangle has an equal length.
FH- =7x-2,
EI-20, and
mZIFG=39°,
Length width equals the area of a rectangle. Therefore, breadth is equal to the product of the area and the length, which is either 20 units or 39 square units.
You need to know a box's height, breadth, and depth in order to determine its volume. These three dimensions may be multiplied to determine the volume.
if < f = 39 degree
so
<G = 39 degree
then
<FIG = 39 + 39 + x = 180 degree
180 - 78 = 102 degree
so <FIG = <EIH
The value of x and m<EIH is 102 degree
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HELP ME ANSWER THIS PLEASE!!!!!!!!!!!!!
The average height of a tree whose age is 25 years old is 364 inches.
What is a linear model?A linear model is an equation that describes a relationship between two quantities that show a constant rate of change.
Given that, a table giving a relation between the height and age of trees.
The linear model representing the same is given by,
y = 13.7x + 21.7, where y is the height and x is the age of a tree,
We need to find the average height of a tree whose age is 25 years old,
To find this, put x = 25 in the linear model given,
y = 13.7 × 25 + 21.7
y = 342.5 + 21.7
y = 364.2
y ≈ 364
Hence, the average height of a tree whose age is 25 years old is 364 inches.
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write a relation to show how bens savings are related to the number of hours he baby sits
Answer:
Ben's savings (S) can be related to the number of hours he babysits (h) by the equation:
S = r * h
where r is the rate of pay per hour that Ben receives for babysitting. This equation shows that Ben's savings increase linearly with the number of hours he babysits, as long as the rate of pay remains constant.
5y⁴-5y³-30y² I need the explanation
Of how to work it out
And the answer
Answer:
20-15-60 is the answer to the question
v=1\10u compared to y=mx+b
By the compare to y =mx + b, the value of slope (m) is 1/10 and value of y - intercept is 0.
What is Equation of line?The equation of line in point-slope form passing through the points
(x₁ , y₁) and (x₂, y₂) with slope m is defined as;
⇒ y - y₁ = m (x - x₁)
Where, m = (y₂ - y₁) / (x₂ - x₁)
We have to given that;
The expression is,
⇒ v = 1/10u
Now, We can write as;
⇒ v = 1/10u
⇒ v = 1/10u + 0
By compare with y = mx + b, we get;
⇒ slope (m) = 1/10
⇒ y - intercept (b) = 0
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You deposit $500 in an account. The account earns $60 simple interest in 8 years. What is the annual interest rate?
The annual interest rate of the deposit is 1.5% per annum.
What is simple interest?Borrowers must pay lenders simple interest as a fee in exchange for a loan. Compound interest is excluded from the calculation and just the original principal is used. Simple interest applies to all loans, not just specific ones. Additionally, it is the kind of interest that banks give their customers on their savings accounts.
Given amount deposit = $500
interest earned in 8 years = $60
time = 8 years
simple interest is given by,
SI = (Prt)/100
P = principle = $500
r = rate per annum
t = time = 8 years
60 = (500*8*r)/100
60*100 = 4000r
r = 6000/4000
r = 1.5% per annum
Hence rate is 1.5% per annum.
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Approximately how many times does an average human heart beat in a lifetime?O 3 x 10^11O 3 x 10^10O 3x 10^9O 3 x 10^8O 3 x 10^7
The approximate number of time does an average human heart beat in a lifetime is 3 x 10⁹
The term called average in math is defined as the mean value which is equal to the ratio of the sum of the number of a given set of values to the total number of values present in the set.
Here we need to find the number of time does an average human heart beat in a lifetime.
As we all know that a normal heart rate when you're not being active is generally between 60 – 100 beats per minute.
And this is called your resting heart rate.
Based on this we have calculated as average of 80 beats per minute, your heart beats about 4,800 times per hour. That's a whopping 115,200 times per day.
Then during an average lifetime, then the human heart will beat more than 2.5 billion times.
When we round off the value then we get approximately 3 billon.
In a scientific notation it can be written as,
=> 3 x 10⁹
Therefore option (c) is correct.
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Using a graphical method, solve the simultaneous equations
X + y = 4
and
y = 2x+ 1
Answer:
x= 1, y=3
Step-by-step explanation:
x + (2x+1)= 4
3x+1=4
3x=4-1
x=1
Take x as 1
1 + y =4
y=4-1
y=3
Is this expression a polynomial? 2x^4 y + 4x^2 y^2
Answer: yes
2x^2yx(x^2+2y)
Step-by-step explanation:
2x^4y+4x^2y^2
2x^2yx(x^2+2y)
find the equation perpendicular to y=-2x+10 through (10, 7)
Answer: y=-2x
Step-by-step explanation:
take formula (y-y1)=m(x-x1), m is slope
plug in x and y values and slope, (y-7)=-2(x-10)
distribute, y-7=-2x+20
y=-2x+27
what type of triangle has the ratio of 1:1:4
Answer:
An isosceles triangle because it will have two side lengths that are equal.
Step-by-step explanation:
Estimate to the nearest tenth. Then locate on a number line.
A can of vegetables has radius 21 in. and height 6.1 in. Find the volume of the can. Use 3.14 for .
Answer:
V = 8446.9 in³
Step-by-step explanation:
Just use the cylinder formula.
V = πr²h
V = (3.14)(21)²(6.1)
V = 8446.9 in³
The surface area of a cylinder is 48π m². the distance around the base of the cylinder is 5π meters and the diameter of a base of the cylinder is 8 meters. what is the height of the cylinder? enter your answer, as a simplified fraction, in the box.
Therefore , the solution of the given problem of fraction comes out to be
height h = 3 meters.
What is meant by fraction?Part of a whole is a fraction. The quantity is written as a quotient in mathematics, where the numerator and denominator are divided. Each is an integer in a simple fraction.
Here,
Given :
=> 48π m².
=> 5π meter and radius = 8/2 = 4 meter
=>π(4)(4)h =48π
=> 16πh = 48π
=> h = 3 meters
Whether it is in the numerator or denominator, a complex fraction contains a fraction. The numerator and denominator of a correct fraction are opposite each other.
Therefore , the solution of the given problem of fraction comes out to be
height h = 3 meters.
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find the solution of the given initial value problem y'' y = f(t), y(0) = 7, y'(0) = 4 where f(t) = 1, 0
The solution of the given initial value problem y'' + y = f(t), y(0) = 7, y'(0) = 4 where f(t) = [tex]\left\{\begin{matrix}1, & 0\leq t\leq \frac{\pi}{2}\\ 0, & \frac{\pi}{2}\leq t < \infty\end{matrix}\right.[/tex] is 1 + 6cost + 4sint + u(t - π/2)(cos(t - π/2) - 1).
The initial value problem y'' + y = f(t), y(0) = 7, y'(0) = 6 where
f(t) = [tex]\left\{\begin{matrix}1, & 0\leq t\leq \frac{\pi}{2}\\ 0, & \frac{\pi}{2}\leq t < \infty\end{matrix}\right.[/tex]
Now f(t) = 1[u(t-0) - u(t-π/2)]
f(t) = u(t) - u(t-π/2)
Laplace transformation of f(t) is
L{f(t)} = L{u(t)} - L{u(t-π/2)}
L{f(t)} = 1/S - [tex]e^{\pi S/2}[/tex]/S
Now taking the Laplace transformation on both side to the equation y'' + y = f(t).
L{y''} + L{y} = L{f(t)}
S^2Y(S) - SY(0) - Y'(0) + Y(S) = 1/S - [tex]e^{\pi S/2}[/tex]/S
S^2Y(S) - 7S - 4 +Y(S) = 1/S - [tex]e^{\pi S/2}[/tex]/S
(S^2 + 1)Y(S) - 7S - 4 = 1/S - [tex]e^{\pi S/2}[/tex]/S
(S^2 + 1)Y(S) = 1/S - [tex]e^{\pi S/2}[/tex]/S + 7S + 4
Divide by (S^2 + 1) on both side, we get
[tex]Y(S) = \frac{1}{S(S^2 + 1)} -\frac{e^{\pi S/2}}{S(S^2 + 1)} + \frac{7S}{(S^2 + 1)}+\frac{4}{S^2 + 1}[/tex]
Simplify
[tex]Y(S) = \frac{1}{S}-\frac{S}{S^2 + 1} -\left[\frac{1}{S}-\frac{S}{S^2 + 1}\right]e^{\pi S/2} + \frac{7S}{(S^2 + 1)}+\frac{4}{S^2 + 1}[/tex]
Now Y(t) = [tex]L^{-1}(Y(S))[/tex]
Y(t) = [tex]L^{-1}\left[\frac{1}{S}\right]-L^{-1}\left[\frac{S}{S^2 + 1}\right] -L^{-1}\left[\frac{e^{\pi S/2}}{S}\right]-L^{-1}\left[\frac{Se^{\pi S/2}}{S^2 + 1}\right] + L^{-1}\left[\frac{7S}{(S^2 + 1)}\right]+L^{-1}\left[\frac{4}{S^2 + 1}\right][/tex]
Y(t) = 1 - cost - u(t - π/2) + cos(t - π/2) u(t - π/2) + 7cost + 4sint
Y(t) = 1 + 6cost + 4sint + u(t - π/2)(cos(t - π/2) - 1)
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The complete question is:
Find the solution of the given initial value problem y'' + y = f(t), y(0) = 7, y'(0) = 4 where f(t) = [tex]\left\{\begin{matrix}1, & 0\leq t\leq \frac{\pi}{2}\\ 0, & \frac{\pi}{2}\leq t < \infty\end{matrix}\right.[/tex]
If x=1 and y=2, what can be said about z if z=xy?
a. If z = xy, then x = 1.
b. If z = xy, then x = 2.
c. If x = 1 and y = 2, then x = zy
d. If z = xy, then z = 3.
e. If z = xy, then z = 2.
Answer: e is the correct option.
Step-by-step explanation:
Option e If z = xy, then z = 2.
What is variable?In Maths, a variable is an alphabet or term that represents an unknown number or unknown value or unknown quantity. The variables are specially used in the case of algebraic expression or algebra. For example, x+9=4 is a linear equation where x is a variable, where 9 and 4 are constants.In the equation z=xy, we can calculate the value of z if we know the values of the other two variables, x and y We're given these two values in the problem, so we can substitute them in. Then, we can perform the overall calculation with the numbers we put in. This will give us our final result.Replacing x with 1 and y with 2z=xy=(1)(2)=2
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there are 5 blue chips and 3 yellow chips in a bag. one chip is drawn from the bag. that chip is placed back into the bag. a second chip is then drawn. what is the probability that the two selected chips are of different colors? express your answer as a common fraction.
The probability that the two selected chips are of different colors is 15/32
How to determine the probabilitiesFrom the question, we have the following parameters that can be used in our computation:
Blue = 5
Yellow = 3
This means that
Total = 5 + 3
Total = 8
The probability of different color is then calculated as
P(Different) = Yellow and Blue or Blue and Yellow
So, we have
P = 3/8 * 5/8 * 2
Evaluate
P = 30/64
Simplify
P = 15/32
Hence, the probability is 15/32
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Bridgette swims several times a week in a lake near her house. This past summer, the average water temperature was 21 °C, which is 5% warmer than it was last summer. What was last summer's average water temperature?
The required average temperature of the last summer is given as 22.10 °C.
The percentage is the ratio of the composition of matter to the overall composition of matter multiplied by 100.
Here,
Let the temperature of last summer's average be x insert,
According to the question,
21 = x - 5%x
x = 21 / 0.95
x = 22.10 °C
Thus, the required average temperature of the last summer is given as 22.10 °C.
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for every 3 crayons, there are 8 blue crayons in john’s backpack. if there were 132 crayons in his backpack, how many are blue?
If there were 132 crayons in John's backpack, the number of blue crayons is equal to blue crayons.
What is a proportion?In Mathematics, a proportion simply refers to an equation which is typically used to represent the equality of two (2) ratios. This ultimately implies that, proportions can be used to establish that two (2) ratios are equivalent and solve for all unknown quantities.
By applying direct proportion to the given information, we have the following mathematical expression:
3 crayons = 8 blue crayons
132 crayons = X blue crayons
By cross-multiplying, we have the following:
3x = 132 × 8
3x = 1,056
x = 1,056/3
x = 352 blue crayons.
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suppose that m is odd. what integer between 1 and m − 1 equals 2−1 mod m?
That m is odd, then the integers between 1 and m − 1 equals 2−1 mod m it must be k=1 and n = 1+m2.
Given an integer m, two integers a and b are congruent modulo m if
m| (a − b).
We write a ≡ b (mod m).
Zero, a positive natural number, or a negative integer denoted by a minus sign are all examples of integers. The inverse additives of the equivalent positive numbers are the negative numbers.
I will also sometimes say equivalent modulo m. Notation note: we are using that "mod" symbol in two different ways.
Let n be the inverse of 2( mod m)
Where 1<n<m−1
2n≡1 ( mod m)
⇒ 2n=1+mk
Since m is odd then m=2l+1
⇒ 2n=1+(2l+1)k
Then solve for n(mod m) as that would put it within 1<n<m−1.
As an aside, we only care for 1≤n≤m−1, we don't care about strict inequalities. Indeed, if m=3 then you'd have a problem as there aren't even any integers strictly between 1 and 3−1
Supposing that m = [tex]2l+1[/tex] then [tex]2*(l+1)[/tex] = [tex]2l + 2[/tex] = m+1
We want integer n=1+mk2 between 1 and m−1.
Since m is odd, we need k to be odd, in order for 1+mk to be divisible by 2 so n is an integer.
If k≥3 then 1+mk2>m−1.
If k≤−1 then 1+mk2<1.
So it must be k=1 and n = 1+m2.
Therefore,
That m is odd, then the integers between 1 and m − 1 equals 2−1 mod m it must be k=1 and n = 1+m2.
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