The given relation is P (|Xn - μ| / σ < 0.25) ≥ 0.99. We need to determine the smallest value of n that satisfies the given relation using the central limit theorem.
Step 1: We know that the standard normal distribution is used in the central limit theorem to approximate the distribution of the sample means. The standard normal distribution has a mean of zero and a standard deviation of one. We need to standardize the given relation so that we can use the standard normal distribution.
Step 2: We substitute the values from the given relation and simplify as follows: P(|Xn - μ| / σ < 0.25) ≥ 0.99P(|Xn - μ| / (o/√n) < 0.25) ≥ 0.99P((Xn - μ) / (o/√n) < 0.25) ≥ 0.99P(-0.25 < (Xn - μ) / (o/√n) < 0.25) ≥ 0.99P(Z < 0.25√n) - P(Z < -0.25√n) ≥ 0.99where Z is the standard normal random variable.
Step 3: We look up the values of -0.25√n and 0.25√n from the standard normal distribution table and find their difference. We use the absolute value of the difference since we are dealing with probabilities. We get: |P (Z < 0.25√n) - P (Z < -0.25√n) | = 0.99. Since the standard normal distribution is symmetric, we have P (Z < 0.25√n) - P (Z > 0.25√n) = 0.99We can rearrange this as: P (Z < 0.25√n) = 0.995P(Z > 0.25√n) = 0.005
Step 4: We look up the value of 0.995 from the standard normal distribution table and find its corresponding z-score. We get: z = 2.58. Using the z-score formula, we can solve for the value of 0.25√n. We get:2.58 = 0.25√nn = (2.58 / 0.25) ²n ≈ 107.58We round up to the nearest integer to get n = 108. Therefore, the smallest number of items n that must be taken in order to satisfy the given relation is 108.
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Marlene went shopping and bought a bunch of candy for her swim team. She bought packs of Skittles for $1.50 each and packs of M&Ms for $2 each. She spent a total of $39 and bought 24 items. How many of each item did she buy?
Answer:
18 Skittles
6 M&Ms
Step-by-step explanation:
Set up an equation:
Variable x = number of skittles
Variable y = number of M&Ms
1.50x + 2y = 39
x + y = 24
In the second equation, isolate a variable:
x = 24 - y
Substitute the value of x for 24 - y in the first equation:
1.50(24 - y) + 2y = 39
Use distributive property
36 - 1.5y + 2y = 39
Combine like terms
36 + 0.5y = 39
Isolate variable y:
0.5y = 3
y = 6
Substitute the value of y for 6 in the second equation:
x + 6 = 24
Isolate variable x:
x = 18
Plug these values into any equation of your choice to see if these values are correct (I'll do both equations just to prove it):
1.50(18) + 2(6) = 39
27 + 12 = 39
39 = 39
Correct
x + y = 24
18 + 6 = 24
24 = 24
Correct
kelly deposits $3,000 at a rate of 5.2ompounded quarterly. with no additional deposits or withdrawals, what is the account balance after 10 years?
The account balance after 10 years will be approximately $5,259.99.
The balance in Kelly's account after 10 years can be determined using the formula for compound interest which is given as:
A = [tex]P(1 + r/n)^{nt}[/tex]
Where, A is the balance after time t, P is the principal amount (initial deposit), r is the annual interest rate (as a decimal), n is the number of times the interest is compounded per year, and t is the time in years.
Using this formula and substituting the given values, we get:
A = [tex]3000(1 + 0.052/4)^{4 \times 10}[/tex]
= [tex]3000(1.0125)^{40}[/tex] ≈ $5,259.99
Therefore, the account balance after 10 years will be approximately $5,259.99.
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sum of -10°C,24°C,-12°C,8°C,-1°C
Answer:
9°C
Step-by-step explanation:
Group the terms and get [tex]24+8-1-10-12[/tex]
Then simplify to get [tex]32-23[/tex]
Subtract and you get 9
So the answer is 9°C.
Hope this helps!
Determine the area of a circle whose radius is 15 feet. Use pie=3.14
Answer:
Down below
Step-by-step explanation:
[tex]A=\pi r^2\\A=3.14*(15)^2\\A=3.14*225\\A=706.5[/tex]
use appropriate algebra and theorem 7.2.1 to find the given inverse laplace transform. (write your answer as a function of t.)
The inverse Laplace transform of X(s) = [tex]1/s^7[/tex] is f(t) =[tex]t^6.[/tex]
To locate the inverse Laplace rework of X(s) =[tex]1/(s^7[/tex]), we are able to use algebraic manipulation and the inverse Laplace transform assets said in Theorem 7.2.1, which permits us to discover the original characteristic while given its Laplace rework.
Using the assets of the Laplace transform, we will rewrite the given expression as:
X(s) = [tex]x^-1(1/s^7) = (1/s^7)[/tex]
We need to locate the feature f(t) such that its Laplace transform is X(s) = [tex]1/s^7[/tex]. By making use of Theorem 7.2.1, we understand that the inverse Laplace remodels of X(s) will give us f(t).
Now, we need to find a characteristic f(t) that has a Laplace transform [tex]1/s^7[/tex]. By examining the Laplace transform a desk or the usage of regarded formulas, we will decide that the Laplace remodel of [tex]t^n[/tex](wherein n is a high-quality integer) is given by means of[tex]n!/s^(n+1).[/tex]
In our case, we're looking for a function whose Laplace remodel is[tex]1/s^7.[/tex]Comparing this with the Laplace transform formulation cited earlier, we see that the exponent within the denominator of sought to be [tex]8 ^(7+1).[/tex]
Hence, f(t) must be t^6 (given that 6+1 = 7), and its Laplace remodel maybe [tex]6!/s^7 = 720/s^7.[/tex]
Therefore, the inverse Laplace transform of X(s) = [tex]1/s^7 is f(t) = t^6.[/tex]
In precis, by applying algebraic manipulation and making use of the inverse Laplace rework assets, we determined that the inverse Laplace transform of [tex]1/s^7 is f(t) = t^6[/tex]. This approach that a unique feature corresponding to the given Laplace rework is [tex]t^6.[/tex].
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is 12, -15, -18, -21 arithmetic
The question is in the photo above plz answer
Answer:
surface area of cuboid=2(lb+bh+hl)
surface area of cuboid=2(12×5+5×2+2×12 )
surface area of cuboid=188in²
iGive a combinatorial proof of 1, 2+ 2.3+3.4++ (x - 1)^n = 2 =>("") Hint: Classify sets of three numbers from the integer interval [0...n] by their maximum element.
By classifying sets of three numbers from the interval [0...n] by their maximum element, we have provided a combinatorial proof of the identity 1, 2 + 2.3 + 3.4 + ... + (x - 1)^n = 2^(n+1).
The combinatorial proof of the identity 1, 2 + 2.3 + 3.4 + ... + (x - 1)^n = 2^(n+1) revolves around classifying sets of three numbers from the integer interval [0...n] by their maximum element.
Let's consider the right-hand side of the equation, which is 2^(n+1). This represents the number of subsets of an n-element set. We can think of each element in the set as having two choices: either it is included in a subset or not. Therefore, there are 2 choices for each element, resulting in a total of 2^(n+1) subsets.
Now, let's look at the left-hand side of the equation, which is the sum 1 + 2 + 2.3 + 3.4 + ... + (x - 1)^n. We can interpret each term as follows:
1: Represents the number of subsets with a maximum element of 0, which is only the empty set.
2: Represents the number of subsets with a maximum element of 1, which includes the subsets {0} and {1}.
2.3: Represents the number of subsets with a maximum element of 2, which includes the subsets {0, 1}, {0, 2}, and {1, 2}.
Similarly, for each subsequent term (x - 1)^n, it represents the number of subsets with a maximum element of x-1.
Now, if we add up all these terms, we are essentially counting the total number of subsets from the original set. This matches the right-hand side of the equation, which is 2^(n+1).
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If the true means of the k populations are equal in an ANOVA model, then MSTR/MSE should be: a. more than 1.00 b. close to 1.00 c. close to 0.00 d. close to -1.00 e. a negative value between 0 and - 1 f. not enough information to make a decision
The correct answer is b. close to 1.00. Ratio close to 1.00 indicates that the between-group variation is similar to the within-group variation.
In an ANOVA (Analysis of Variance) model, MSTR refers to the mean square treatment (or between-group variation), while MSE refers to the mean square error (or within-group variation).
If the true means of the k populations are equal, it means that the between-group variation is similar to the within-group variation, and there is no significant difference between the group means.
In this scenario, we would expect the MSTR/MSE ratio to be close to 1.00 (answer b). A ratio close to 1.00 indicates that the between-group variation is similar to the within-group variation, supporting the assumption that the true means of the populations are equal.
Therefore, the correct answer is b. close to 1.00.
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Tue diameter of a bike is 27 inches if the wheel makes 15 complete rotations how far does the bike travel?
Answer:
1271.7 inches
Step-by-step explanation:
Given
[tex]d = 27in[/tex]
[tex]Rotations = 15[/tex]
Required
The total distance moved
First, we calculate the circumference of the wheel
[tex]c = \pi *d[/tex]
[tex]c = 3.14 * 27[/tex]
[tex]c = 84.78[/tex]
This represents the distance traveled in one complete rotation.
So, in 15 rotations.
[tex]Total = 15 * 84.78[/tex]
[tex]Total = 1271.7[/tex]
What is the value of g^-1(7)? PLEASE HELP!!! I’ll give brainliest!!
Answer: g-7
Step-by-step explanation:
Answer:
Step-by-step explanation:
5 is the correct answer
The number of calories of a food item varies directly with the size of the portion. If a 2-inch slice of a certain delicacy contains 170 calories, how many calories are in a 3-inch slice?
Answer:
A 3-inch slice of the food item contains 255 calories.
Step-by-step explanation:
Set calories per inch slice as variable x.
2x = 170
x = 85 cal
Since each inch slice of the food item contains 85 calories, a 3-inch slice would contain:
3x = 3(85) = 255 cal
what is 25/x 15/30 can you please help
Answer:
x = 50.... 25/50 = 15/30 or 1/2 = 1/2
Step-by-step explanation:
25/x = 15/30
consider the cross multiply
25 * 30 = 15 * x
750 = 15x
divide both sides with 15 to make the co-efficient of x, 1.
x = 50
Answer:
5×15/6
5×5/2
25/2
12.2 is your answer ☺️☺️☺️. If I'm right so,
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whats 38 divided by 70
Answer:
.5428571 repeating
or 19/35
Find the compound interest paid at the end of 11 years and 33 months when a sum of ₹20000 is invested at a rate of 6% per annum compounded annually?
DONT ADD LINKS PLEASE ANSWER CORRECTLY AND FAST, PLEASE I WILL MARK YOU BRAINLIEST FOR CORRECT ANSWER
Answer:
The total would be 36,600.
Step-by-step explanation:
First you will convert 33 months into years which is 2 years and 10 months.
You will then add the 2 years and the 11 years together which gets you 13 so now you have a total of 13 years and 10 months. (this information will be used later)
Now we will start by multiplying the 20000 by 16% and we should get a total of 1200. That means the interest is 1200 per year so lets first take our 13 years and multiply it by the 1200 (13x1200). You should have 15600 as an answer (save this number for later) since we have 10 months were going to take that 1200 and divide it into 12 month meaning the monthly interest should be 100 per month since we have 10 months we will multiply 100 by 10 and get 1000. Now lets bring back that 15600 and add the additional 1000 to it, our answer should be 16,600, and then we add the original 20000 to the 16600 and your final answer should be 36600.
Does any one know the answer to this thank you
Answer:
The First choice
Step-by-step explanation:
It takes me into 30 minutes to walk from home to school when walking at 5 km per hour What is her average cycling speed If it takes her 15 minutes by bike to travel the same distance
Answer:
10 km/h
Step-by-step explanation:
I'm not really sure about this but no ones answering your question and I wanna help.
So basically to calculate the average speed you need to divide the distance travelled by time taken
But you do not have the distance traveled. But it is mentioned that it takes u 30 minutes to walk from home to school when walking at 5 km/h so to find the distance all you have to do is... 30 x 5 = 150 km
Now that we have the time and distance all we have to do is find the average speed.
Average Speed = distance ÷ time
So 150 ÷ 15 = 10 km/h
Please help.
Indices and Standard Form question
Answer:
A= 4.5×10^11
B= 3.5×10^3
Step-by-step explanation:
A
(5×10^3)= 5000
(9×10^7)=90000000
multiply both of them
=450000000000 or 4.5×10^11
B
(7×10^5)=700000
(2×10^2)=200
700000÷200
3500 or 3.5×10^3
Q1) Eruptions of the Old Faithful geyser in Yellowstone National Park typically last from 1.5 to 5 minutes. Between eruptions are dormant periods, which typically last from 50 to 100 minutes. A dormant period can also be thought of as the waiting time between eruptions. The duration in minutes for 40 consecutive dormant periods are given in the following table. 91 82 84 85 80 73 72 84 86 76 51 70 71 83 79 79 67 76 60 81 55 53 51 53 45 49 67 76 86 88 82 68 82 51 51 75 86 沙575 66 Assuming that the waiting time follows an Exponential distribution with mean parameter A, develop a uniformly most powerful test of size a = 0.01 for Hoλ2 80 vs H₁ A<80. Based on this test, draw a conclusion.
The duration in minutes for 40 consecutive dormant periods are given in the following table:91 82 84 85 80 73 72 84 86 76 51 70 71 83 79 79 67 76 60 81 55 53 51 53 45 49 67 76 86 88 82 68 82 51 51 75 86 575 66.
Assuming that the waiting time follows an exponential distribution with mean parameter A, a uniformly most powerful test of size α = 0.01 for H o λ^2=80 vs H1 A<80 can be developed as follows: The null and alternative hypotheses are as follows:H0:λ^2=80, that is, the mean of the exponential distribution is 80 squared.H1:A<80, which implies that the mean waiting time between eruptions is less than 80 squared.α=0.01 is the level of significance.
The following test statistic T is used: T = [n(λ^2-80)] / 80^2where n is the sample size, and the critical region is the left-tail rejection area. The probability of observing the values in the given sample or a more extreme set of values is calculated as follows: Since we are performing a one-tailed test, we divide α by 2.α/2 = 0.005
The area in the left tail is 0.005, and the corresponding z-score is -2.33.The null hypothesis is rejected if the computed value of the test statistic falls in the critical region, which is in the left-tail rejection region. T < -2.33
Since the test statistic T = -1.91 falls in the non-critical region, we fail to reject the null hypothesis at the α=0.01 level of significance. Therefore, based on this test, we can conclude that there is insufficient evidence to suggest that the mean waiting time between eruptions is less than 80 squared.
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Find the general solution of the given differential equation. 7 dy dx + 56y = 8
y(x) =
Give the largest interval I over which the general solution is defined. (Think about the implications of any singular points. Enter your answer using interval notation.)
Determine whether there are any transient terms in the general solution.
The general solution of the given differential equation 7dy/dx + 56y = 8 is y(x) = -x/8 + C e^(-8x/7), where C is a constant.
To solve the differential equation, we first rearrange it to isolate dy/dx: dy/dx = (8 - 56y)/7. This is a first-order linear differential equation. The integrating factor is e^(∫(-56/7)dx) = e^(-8x/7). Multiplying both sides of the equation by this integrating factor, we obtain e^(-8x/7) dy/dx + 8e^(-8x/7)y = 8e^(-8x/7). The left-hand side can be written as the derivative of y multiplied by e^(-8x/7). Integrating both sides gives ∫d(y e^(-8x/7)) = ∫8e^(-8x/7) dx. Solving these integrals and rearranging, we find the general solution y(x) = -x/8 + C e^(-8x/7), where C is the constant of integration.
The largest interval I over which the general solution is defined is (-∞, ∞) since there are no singular points or restrictions mentioned in the differential equation. This means that the solution is valid for all real values of x.
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Assume that a sample is used to estimate a population proportion μ. Find the margin of error M.E. that corresponds to a sample of size 10 with a mean of 33.7 and a standard deviation of 13.3 at a confidence level of 95%. Report ME accurate to one decimal place because the sample statistics are presented with this accuracy. M.E. = _________ Answer should be obtained without any preliminary rounding. However, the critical value may be rounded to 3 decimal places.
The margin of error for a 95% confidence level is 8.2.
How to find the margin of error?The margin of error (ME) is determined using the formula:
ME= z ∗ σ/√n
where:
z is the z-score for the desired confidence level
σ is the population standard deviation
n is the sample size
For a 95% confidence level, the z-score is 1.96. Thus, we have:
z = 1.96
σ = 13.3
n = 10
Substituting these values into the formula, we have:
ME = 1.96 ∗ 13.3/√10
ME = 8.2
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Is (2,0) a solution for the equation -3+9y=-6
Answer:
[tex]y=-\frac{1}{3}[/tex]
Step-by-step explanation:
- 3 + 9y = - 6
- 3 + 3 + 9y = - 6 + 3
9y = - 3
9y ÷ 9 = - 3 ÷ 9
[tex]y=- \frac{1}{3}[/tex]
The table below gives a record of variations of the values of y with the values of x. Draw a scatter plot for the data.
x
0.4
1.2
2.0
3.1
4.5
5.7
7.1
8.4
9.3
9.8
y
7.8
7.1
6.8
6.0
5.2
4.3
3.4
2.3
1.1
0.5
a.
On a graph, points are at (2, 6.9), (9.3, 1.2), (9.8, 0).
c.
On a graph, points are at (0.4, 7.8), (3.1, 6.0), and (9.8, 0.5).
b.
On a graph, points are at (2, 7), (9.3, 1.2), (9.8, 1.5).
d.
On a graph, points are at (1.2, 7.2), (9.3, 1.2), (9.8, 0.2).
Please select the best answer from the choices provided
A
B
C
D
Answer: taake this link, it has all the answers
Step-by-step explanation: https://quizlet.com/183183758/statistical-studies-scatterplots-practiceamdm-flash-cards/
The best option for the points on the graph is points are at (0.4, 7.8), (3.1, 6.0), and (9.8, 0.5).
What are co ordinate axis?In two-dimensional Cartesian geometry, two intersecting straight lines are used as reference lines. In three-dimensional Cartesian geometry, three straight lines with a common point are the intersections of the three coordinate reference planes.
Estimation of the coordinates from the graph:From the attached file of scatter plot for the data, it is clear that the for every value of x there is a suitable value of y which was given in the question.
Considering x values and plot the y vale on the graph.
For x = 0.4; y = 7.8
For x = 3.1; y = 6.0
and for x = 9.8; y = 0.5
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A sample of two items is selected without replacement from a batch. Describe the ordered sample space for the following batch:
(a)The batch contains 3 defective items and 10 good times.Hint: suppose we denote defective item by ‘d’ and good item as ‘g’, so one possible outcome could be "dg".
(b)The batch contains the items {a, b, c, d}.
For both scenarios, a sample is selected without replacement from a batch of items. In the first scenario, the batch contains 3 defective items ('d') and 10 good items ('g'). The ordered sample space consists of all possible ordered pairs of items: {dd, dg, gd, gg}. In the second scenario, the batch contains the items {a, b, c, d}. The ordered sample space also consists of all possible ordered pairs of items: {aa, ab, ac, ad, ba, bb, bc, bd, ca, cb, cc, cd, da, db, dc, dd}.
In the first scenario, the ordered sample space is derived by considering all possible combinations of the two items selected from the batch. Since the selection is done without replacement, the first item can be either defective ('d') or good ('g'). For each case, the second item can also be defective or good, depending on what was chosen as the first item. Therefore, the ordered sample space consists of four possibilities: dd, dg, gd, and gg.
In the second scenario, the batch consists of four distinct items: a, b, c, and d. Again, the ordered sample space is obtained by considering all possible combinations of the two items selected without replacement. Since there are four items, there are 16 possible combinations. Each combination is represented by an ordered pair of the selected items, such as aa, ab, ac, and so on.
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pls help i'll give u a brainlyy
Answer: -633
Explanation:
formula is a0= a1 + (n-1)d
a0 is what ur tryig to find
a1 = -14
n= 60
d = -11
Answer:
-674Step-by-step explanation:
As,
There is always a gap of -11 in all the proceedings
Hence,
- 14 -11 = -25
-14 -(11 ×2) = -36
So,
60th term
-14 -(11 × 60) = -674 (Ans)
The volume of a cylinder can be determined using the formula V=πr2h, where r and h represent the radius and height of the cylinder, respectively. A volume of paint expressed as (8x3 + 31x2 + 32x)π and a volume of paint expressed as (10x3 + 17x2)π are poured into a paint can in the shape of a cylinder. Determine possible expressions for the radius of the can and the depth of the paint in the can.
Answer:
Possible expressions for the radius of the can and the depth of the paint in the can are [tex]r = \sqrt{9\cdot x^{2}+24\cdot x+16}[/tex] and [tex]h = 2\cdot x[/tex], respectively.
Step-by-step explanation:
Let be the initial volumes of the initial cans represented by these expressions:
[tex]V_{1} = (8\cdot x^{3}+31\cdot x^{2}+32\cdot x)\cdot \pi[/tex] (1)
[tex]V_{2} = (10\cdot x^{3}+17\cdot x^{2})\cdot \pi[/tex] (2)
The resulting volume of the paint can is the sum of the two functions:
[tex]V_{3} = (18\cdot x^{3}+48\cdot x^{2}+32\cdot x)\cdot \pi[/tex] (3)
Then, we proceed to factor the polynomial:
[tex]V_{3} = 2\cdot (9\cdot x^{2}+24\cdot x +16)\cdot x \cdot \pi[/tex]
[tex]V_{3} = \pi\cdot (9\cdot x^{2}+24\cdot x + 16)\cdot (2\cdot x)[/tex] (3b)
By direct comparison with the volume formula for the cylinder we have the following expressions:
[tex]r^{2} = 9\cdot x^{2}+24\cdot x + 16[/tex]
[tex]r = \sqrt{9\cdot x^{2}+24\cdot x+16}[/tex]
[tex]h = 2\cdot x[/tex]
Possible expressions for the radius of the can and the depth of the paint in the can are [tex]r = \sqrt{9\cdot x^{2}+24\cdot x+16}[/tex] and [tex]h = 2\cdot x[/tex], respectively.
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Translate the description as an algebraic expression:
double the product of 15 and k
Answer:
2•15k
Step-by-step explanation:
Answer:
2(15k)
Step-by-step explanation:
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use the inner product u, v = 2u1v1 u2v2 in r2 and the gram-schmidt orthonormalization process to transform {(2, 1), (−2, −5)} into an orthonormal basis.
The orthonormal basis for (2, 1), (2, 5) is therefore u1, u2 = (2/5, 1/5), (2/5, -1/5) because u2 = v2_orth/||v2_orth|| = (2/5, -1/5).
In R2, the internal result of the two vectors u and v is as follows: The Gram-Schmidt procedure can be used to request the transformation of (2, 1), (2, 5) into an orthonormal premise. u, v = 2u1v1 + u2v2. An orthonormal premise is made by changing over a bunch of directly free vectors utilizing the Gram-Schmidt process. Our set's principal vector, v1 = (2, 1), should serve as our starting point.
We standardize v1 to obtain our first orthonormal premise vector: We must locate the second vector in our set, v2 = (-2, -5), and we can orthogonalize v2 by deducting its projection from u1: u1 = v1/||v1|| = (2/5, 1/5) proj_u1(v2) = (v2 u1)u1 = (- 8/5, - 4/5)v2_orth = v2 - proj_u1(v2) = (6, - 21/5)Our second orthonormal premise vector is acquired by normalizing v2_orth: The orthonormal reason for (2, 1), (2, 5) is subsequently u1, u2 = (2/5, 1/5), (2/5, - 1/5) in light of the fact that u2 = v2_orth/||v2_orth|| = (2/5, - 1/5).
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A test for divisibility by 11 is to see if the digits taken in order and alternately added and subtracted produce a number which is divisible by 11. Consider 5-digit numbers of the form "abcde'. Show that "abcde' will be divisible by 11 if and only if a - b+c-d+e is divisible by 11.
An test for divisibility that a 5-digit number "abcde" will be divisible by 11 if and only if "a - b + c - d + e" is divisible by 11.
To show that a 5-digit number "abcde" is divisible by 11 if and only if "a - b + c - d + e" is divisible by 11, the concept of modular arithmetic.
The 5-digit number "abcde" as a sum of its digits multiplied by their respective place values:
"abcde" = a × 10000 + b × 1000 + c × 100 + d × 10 + e
Then express "a - b + c - d + e" in terms of the digits:
a - b + c - d + e = a × (10,000 mod 11) - b × (1,000 mod 11) + c × (100 mod 11) - d × (10 mod 11) + e
examine the patterns of the modulos:
10,000 mod 11 = 1
1,000 mod 11 = 10
100 mod 11 = 1
10 mod 11 = 10
Substituting these values back into the expression,
a - b + c - d + e = a × 1 - b ×10 + c × 1 - d × 10 + e
Simplifying further:
a - b + c - d + e = a - b + c - d + e
observe that the expression "a - b + c - d + e" is equivalent to the original 5-digit number "abcde." This means that if "abcde" is divisible by 11, then "a - b + c - d + e" will also be divisible by 11.
Conversely, if "a - b + c - d + e" is divisible by 11, it implies that the expression and the 5-digit number "abcde" have the same remainder when divided by 11. Since they are equivalent, "abcde" must also be divisible by 11.
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