let m be a positive integer. show that a mod m = bmodmifa≡b(modm).
Answers
To show that a mod m = b mod m if a ≡ b (mod m), we can use the definition of modular arithmetic.
To show that a mod m = b mod m if a ≡ b (mod m) for a positive integer m, we can follow these steps:
1. First, understand the given condition: a ≡ b (mod m) means that a and b have the same remainder when divided by the positive integer m. In other words, m divides the difference between a and b. Mathematically, this can be written as m | (a - b), which means there exists an integer k such that a - b = mk.
2. Next, recall the definition of modular arithmetic: a mod m is the remainder when a is divided by m, and similarly, b mod m is the remainder when b is divided by m.
3. We know that a - b = mk, so a = b + mk.
4. Now, divide both sides of the equation a = b + mk by m.
5. Since b and mk are both divisible by m, the remainder of this division will be the same for both sides. In other words, a mod m and b mod m have the same remainder when divided by m.
6. Therefore, we can conclude that a mod m = b mod m if a ≡ b (mod m) for a positive integer m.
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Related Questions
By bike, Martin covers 9 km in 30 minutes. Express this situation as a unit rate.
Answers
Answer:
Step-by-step explanation:
(9 km)/(30 min) × (60 min)/hr = 18 km per hour
Tamar's house is greater than 472 and less than 500 which number can be on Tamar's house?
Answers
490
472 <490<500
it fits the criteria
If z^3= x^3 + y^2, dx/dt=3, dy/dt=2, and z>0, find dz/dt at (x,y) =(4,0). Please give an exact answer. Provide your answer below:
Answers
The rate of change of z with respect to t, dz/dt, at (x,y) = (4,0) is equal to 48, based on the given equations and the differentiation of z³ = x³ + y².
To find dz/dt at (x,y) = (4,0), we need to differentiate the equation z³ = x³ + y² with respect to t.
Differentiating both sides of the equation, we get:
3z² * dz/dt = 3x² * dx/dt + 2y * dy/dt
Given dx/dt = 3 and dy/dt = 2, and since (x,y) = (4,0), we have
3z² * dz/dt = 3(4)² * 3 + 2(0) * 2
3z² * dz/dt = 3(16) * 3
3z² * dz/dt = 144z²
Now, let's solve for dz/dt
dz/dt = (144z²) / (3z²)
dz/dt = 48
Therefore, dz/dt at (x,y) = (4,0) is equal to 48.
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identify points of discontinuity and classify the type of each discontinuity
Answers
Infinite discontinuity
Explanations:The given equation is:
\(y\text{ = }\frac{2x}{x+1}\)The function will be undefined at the point x + 1 = 0
x = -1
This means that x = -1 is a point of discontinuity
Substituting x = -1 into the equation:
\(\begin{gathered} y\text{ = }\frac{2(-1)}{-1+1} \\ y\text{ = }\frac{-2}{0} \\ y\text{ = }\infty \end{gathered}\)This is an infinite discontinuity
A rock thrown into a still pond causes a circular ripple. If the radius of the ripple is increasing at 2 feet per second, how fast is the area changing when the radius is 10 feet
Answers
When the radius of the ripple is 10 feet and increasing at a rate of 2 feet per second, the area of the ripple is changing at a rate of 40π square feet per second.
We can use the formula for the area of a circle to solve this. This formula is given as:
Area of a circle = πr², Where r is the radius of the circle.
To find how fast the area is changing, we can use differentiation with respect to time. This gives us:
dA/dt = 2πr (dr/dt).
From the question, dr/dt = 2 feet per second, and r = 10 feet.
Substituting these values into the above equation gives us:
dA/dt = 2π × 10 × 2 = 40π.
Therefore, the area of the ripple is changing at a rate of 40π square feet per second when the radius is 10 feet.
The answer is 40π.
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calculate the cost of manufacturing a standard cereal box if cardboard costs 0.05 per square inch312 volume286 surface area
Answers
Given:
The cardboard costs 0.05 per square inch
And the surface area = 286
So, the total cost is the product of the surface area and the cost per square in.
Total cost =
\(286*0.05=14.3\)So, the answer will be Total cost = $14.30
you are interested in what factors are related to home prices. you have data on 45 homes and want to jointly test whether the size of the house, number of bedrooms in the house, the number of bathrooms in the house, and the size of the kitchen are related to the price of the house (i.e. the null hypothesis is that none of these factors is related to the price of the house). what is the critical value for this test at a significance level of 5%?
Answers
The critical value for this test at a significance level of 5% is determined by F-distribution with degrees of freedom (df1, df2) critical value for this test at a significance level of 5% is 2.885.
In this case, we are testing the relationship between the price of the house and four factors (size of the house, number of bedrooms, number of bathrooms, size of the kitchen). Since there are four factors, the degrees of freedom for the numerator (df1) is 4, and the degrees of freedom for the denominator (df2) is the total sample size minus the number of factors, which is 45 - 4 = 41.
To find the critical value for the F-test at a significance level of 5%, we can use a statistical table or software. For the given degrees of freedom (4, 41), the critical value is approximately 2.885.
Therefore, the critical value for this test at a significance level of 5% is 2.885. If the calculated F-statistic is greater than this critical value, we would reject the null hypothesis and conclude that at least one of the factors is related to the price of the house.
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In ABC, AB = 5 feet and BC = 3 feet.
Which inequality represent all possible values for the length of AC, in feet?
1. 2 <_AC <_ 8
2. 2 < AC < 8
3. 3 <_ AC<_ 7
4. 3 < AC <7
Answers
Inequalities help us to compare two unequal expressions. The value of AC should be 2≤AC≤8, the correct option is 1.
What are inequalities?Inequalities help us to compare two unequal expressions. Also, it helps us to compare the non-equal expressions so that an equation can be formed.
It is mostly denoted by the symbol <, >, ≤, and ≥.
As per the law of triangle inequality, the sum of any two sides of the triangle must be greater than or equal to the third side, this will give three conditions,
Since the third condition is going to be true for all the whole numbers. Therefore, as per the first two conditions, the value of AC should be 2≤AC≤8.
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Write two fractions equivalent to 9/12) PLEASE HELP I HAVE TOO RETURE THIS BY TOMORROW
Answers
Answer:
3/4, and 6/8
Step-by-step explanation:
3/4 = 9/12 simplfied
6/8 = 3/4 times two
Answer:
3/4 and 18/24
Step-by-step explanation:
I simplified it for the first one and doubled both of the numbers for the second.
I hope this helped.
For the function f(x) = 2x-4/x+3
What is the x-intercept, y-intercept, and vertical asymptotes
Answers
Answer:
x-intercept: (2,0)
y-intercept: (0,-4/3)
vertical asymptote: x = -3
Step-by-step explanation:
To find the x-intercepts, we can set f(x) to 0, which is y=0:
2x-4/x+3 = 0
2x-4 = 0
2x = 4
x = 2
To find the y-intercepts, we can set x to 0:
2(0)-4/(0)+3 = y
y = -4/3
To find the vertical asymptote, we can set the denominator equal to 0:
x+3 = 0
x = -3
A, B and C lie on a straight line segment.
A, E and D lie on a straight line segment.
BE = 1.75m, CD = 7m and BC = 1.5m.
Work out the length of AB.
A
E
B
D
С
Answers
The value of AB id 0.5 unit.
What is AAA similarity ?The Angle-Angle-Angle (AAA) criterion for the similarity of triangles states that “If in two triangles, corresponding angles are equal, then their corresponding sides are in the same ratio (or proportion) and hence the two triangles are similar”.
What are corresponding angles?Any pair of angles each of which is on the same side of one of two lines cut by a transversal and on the same side of the transversal are called corresponding angles and they are always equal.
According to the given question.
A, B and C lie on the straight line segment. and A, E, and D are also lie in the straight line segment.
Also, BE = 1.75m, CD = 7m and BC = 1.5m
Now, in ΔAEB and ΔADC
EB is parallel to BC
So,
∠AEB ≅ ∠ADC (corresponding angles)
∠ABE ≅ ∠ACD (corresponding angles)
∠EAB ≅ ∠DAC (common angle to the both triangle)
Hence, by AAA Similarity rule
ΔAEB is similar to ΔADC
Therefore,
\(\frac{AE}{AD} =\frac{BE}{CD} =\frac{AB}{AC}\) (sides of the similar triangles are in proportion)
So,
\(\frac{BE}{CD} =\frac{AB}{AC} \\\implies \frac{AB}{AC} =\frac{1.75}{7} ....(i)\)
Now, AC = AB + BC (as, A, B, and C lie on a straight line)
\(\implies \frac{AB}{AB+BC} =\frac{1.75}{7}\) (from (i))
\(\implies \frac{AB}{AB+1.5} =\frac{1.75}{7}\)
\(\implies 7AB = 1.75AB + 1.75(1.5)\)
\(\implies 7AB -1.75AB = 2.625\)
\(\implies 5.25AB = 2.625\)
\(\implies AB = \frac{2.625}{5.25} = 0.5\)
Hence, the value of AB id 0.5 unit.
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please answer this.
thank you
Answers
Answer:
C
Step-by-step explanation:
cross multiply
Answer:
2(x + 4)
Step-by-step explanation:
This is because when you open up the brackets and multiply....
2 × x = 2x
and 2 × 4 = +8
So 2(x + 4) = 2x + 8
The bill with the greatest value ever printed in the United States was worth 10 to the 5th power dollars which of the following is equal to the value of the bill
Answers
Answer: 50,000
Step-by-step explanation:
what is the probability that an arrangement of a, b, c, e, f, g begins and ends with a vowel?
Answers
The probability that an arrangement of a, b, c, e, f, g begins and ends with a vowel is 1/ 5, that is 0.2.
Therefore, the answer is 0.2.
In an arrangement of a, b, c, e, f, g, the total number of alphabets is 6 and number of vowels is 2.
The probability that an arrangement of a, b, c, e, f, g begins with a vowel is
P1 = 2/ 6
P1 = 1/ 3
The probability that an arrangement of a, b, c, e, f, g ends with a vowel given it begin with a vowel( In this case the total number of alphabets left is 5 and number of vowels left is 1) is
P = 1/ 5
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please help factorise 7ab+a
Answers
Answer:
= a(7b + 1)
Step-by-step explanation:
7ab + a
a(7b + 1).
the second derivative of a function f is given by f''(x)=x(x-3)^5(x-10)^2
Answers
The second derivative of the function f is expressed as f''(x) = x(x-3)^5(x-10)^2. This information provides insights into the behavior and critical points of the function.
The given expression, f''(x) = x(x-3)^5(x-10)^2, represents the second derivative of a function f with respect to the variable x. The second derivative provides valuable information about the behavior of the function, particularly regarding concavity and inflection points.
The equation indicates that the function has factors of x, (x-3)^5, and (x-10)^2. The term x indicates that the function includes a linear component, while the factors (x-3)^5 and (x-10)^2 suggest that the function may exhibit multiple inflection points and changes in concavity around x = 3 and x = 10.
The expression does not provide information about the original function f(x) or its first derivative f'(x), but it does give valuable insights into the higher-order behavior of the function and can help analyze critical points and concavity characteristics when combined with additional information about the function.
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Enter the correct answer in the box.
Write the inverse of the cube root function used to represent the situation in model 2, .
Answers
The inverse of the cube root function used to represent the situation in model 2 is g(x) = x^3/27.
The cube root function is represented as f(x) = 3√x in model 2. To find the inverse of this function, we can first switch x and y and then solve for y. Therefore, let's start by replacing f(x) with y.
y = 3√x
Now, we'll switch x and y.
x = 3√y
Next, we'll isolate y by cubing both sides of the equation.
x^3 = (3√y)^3
x^3 = 27y
Finally, we'll solve for y by dividing both sides by 27.
y = x^3/27
Therefore, the inverse of the cube root function used to represent the situation in model 2 is g(x) = x^3/27.
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Molly went shopping and her bank account got below $7.50
Answers
How many singlets are expected in the 1h nmr spectrum of 2,2,4,4-tetramethylpentane? 1 4 3 2
Answers
Only one singlet is expected in the 1H NMR spectrum of 2,2,4,4-tetramethylpentane. So, correct option is A.
In the 1H NMR spectrum, the number of singlets corresponds to the number of unique hydrogen environments in the molecule. Each unique hydrogen environment, where the hydrogens are chemically equivalent, will produce a singlet peak.
In 2,2,4,4-tetramethylpentane, all the carbons are identical, and each carbon is bonded to three hydrogen atoms. The four methyl groups (-CH3) are also chemically equivalent to each other. Therefore, all the hydrogens in this molecule are in equivalent environments, and there are no differences between them.
Since all the hydrogens are chemically equivalent, they will produce a single peak in the 1H NMR spectrum. This single peak is called a singlet.
Hence, the correct answer is option a, as only one singlet is expected in the 1H NMR spectrum of 2,2,4,4-tetramethylpentane.
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Answer: Two singlets.
Step-by-step explanation:
The methyl groups are all equivalent with no neighbors, so they all give rise to only 1 peak.
There is a methylene group that gives rise to another peak.
So the total is 2 singlets.
the reliability of a widely used microprocessor is typically expressed as time between fail- ures (tbf). suppose tbf is a right skewed distributed random variable that averages 3,000 cycles with a standard deviation of 54 cycles. 24. suppose the purchaser of these microprocessors will draw a random sample of size 36. what is the mean and standard deviation of the average of the 36 measurements, to two decimal places?
Answers
The mean and the standard deviation for the average of the 36 measurements is given as follows:
Mean of 3000 cycles.Standard deviation of 9 cycles.What is defined by the Central Limit Theorem?The Central Limit Theorem defines the distribution of the sampling distribution of sample means with size n of a normally distributed variable with mean \(\mu\) and standard deviation \(\sigma\).
For the sampling distribution, the mean and the standard deviation are given as follows:
Mean: \(\mu\), that is, the same as the population mean.Standard deviation: \(s = \frac{\sigma}{\sqrt{n}}\), which is the population standard deviation divided by the square root of the sample size.Considering the sample size of n = 36, the standard deviation is obtained as follows:
\(s = \frac{54}{\sqrt{36}} = 9\)
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1/2|x + 8| = |4x − 1|
Answers
K-7d use k=9 and d=-2
Answers
Answer:
-5
Step-by-step explanation:
substitute in the values. 9-7(-2)
-7*-2=14
9-(14)
9-14=
-5
Which conditional probabilities are correct? check all that apply. p(d | f) = startfraction 6 over 34 endfraction p(e | d) = startfraction 7 over 25 endfraction p(d | e) = startfraction 7 over 25 endfraction p(f | e) = startfraction 8 over 18 endfraction p(e | f) = startfraction 13 over 21 endfraction
Answers
The conditional probabilities that are correct are P(D | F) = 6/34, P(E | D) = 7/25 and P(F | E) = 8/18
How to determine the true conditional probabilities ?The formula to compute the conditional probability P(A|B) is:
P(A | B) = P(A and B)/P(B)
The above means that the probability of event A such that the event B has already occurred
When the above formula is applied to the give data in the complete, we have:
P(D | F) = 6/34
P(E | D) = 7/25
P(D | E) = 7/18
P(F | E) = 8/18
P(E | F) = 8/34
Hence, the conditional probabilities that are correct are P(D | F) = 6/34, P(E | D) = 7/25 and P(F | E) = 8/18
See attachment for complete question
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Answer: a,b,d
Step-by-step explanation:
trust
3. (10 points) The following linear programming problem has no feasible solution. min 0x₁ + 0x₂ 3x₁ + 2x₂ ≥ 30 2x1 + x₂ ≥ 40 5x₁ + 3x₂ ≤ 50 X₁, X₂ 20 s. t. (1) (2) Assume that we have • the penalty cost $100 for failing to satisfy 1 unit of 30 in the 1st constraint; the penalty cost $50 for failing to satisfy 1 unit of 40 in the 2nd constraint; also $10 penalty is assessed for each unit exceeding 50 in the 3rd constraint. • Please add some deviational variables on these three constraints to formulate this problem as a goal programming problem to minimize the penalty cost (just write a linear programming model with explanations on new variables).
Answers
To minimize the penalty cost, the linear programming problem is formulated as a goal programming problem by introducing deviation variables (d₁, d₂, d₃) and surplus variables (s₁, s₂), with the objective function Z = 100d₁ + 50d₂ + 10d₃ and the corresponding constraints.
To formulate the given linear programming problem as a goal programming problem to minimize the penalty cost, we can introduce deviation variables on each constraint and assign penalty weights to these variables. The goal is to minimize the total penalty cost:
d₁: Deviation variable for the first constraint (3x₁ + 2x₂ ≥ 30)
d₂: Deviation variable for the second constraint (2x₁ + x₂ ≥ 40)
d₃: Deviation variable for the third constraint (5x₁ + 3x₂ ≤ 50)
We also need to introduce positive surplus variables to ensure the original constraints are not violated:
s₁: Surplus variable for the first constraint (3x₁ + 2x₂ ≥ 30)
s₂: Surplus variable for the second constraint (2x₁ + x₂ ≥ 40)
Now, let's formulate the linear programming model with penalty costs and deviation variables:
Minimize:
Z = 100d₁ + 50d₂ + 10d₃
Subject to:
3x₁ + 2x₂ - d₁ + s₁ = 30
2x₁ + x₂ - d₂ + s₂ = 40
5x₁ + 3x₂ + d₃ ≤ 50
x₁, x₂, d₁, d₂, d₃, s₁, s₂ ≥ 0
The objective function minimizes the penalty cost by summing the products of deviation variables and their respective penalty weights.
The first constraint includes the deviation variable d₁ to measure the deviation from the minimum requirement of 30. The surplus variable s₁ ensures that the original constraint is not violated.
Similarly, the second constraint includes the deviation variable d₂ and the surplus variable s₂.
The third constraint includes the deviation variable d₃ to penalize any excess beyond the upper limit of 50.
All decision variables (x₁, x₂, d₁, d₂, d₃, s₁, s₂) are non-negative.
Solving this goal programming problem will minimize the penalty cost associated with failing to satisfy the original constraints.
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Suppose that a phone that originally sold for $800 and is worth 3/5 of its value each year after it is released. Write an equation for the value of the phone, y years after it is released.
Answers
Answer:
v = 800 - 3/5y
Step-by-step explanation:
v = value
y = years
The original cost/value was 800 dollars which means you would have to subtract the 3/5 decrease in value each year.
v = 800 - 3/5y
Answer:
800/(y*3/5)
Step-by-step explanation:
If f(x) = x2 – 3, then f(-1) is equivalent
to
Answers
Answer:
negative 5. or -5
Step-by-step explanation:
f(-1)= (-1)2-3
2 times -1= -2
-2-3= -5
f= -5
this question is -5
(2x² - 6x + 5) + (7x²-x-9)
need help finding sum or difference
Answers
Answer:
the answer is either x= 4 , or it is 9\(x^{2} \\\) - \(x\) - 10
Step-by-step explanation:
Sorry dude, I'm confused a little bit on what to do, but i think one of these is right.
lmk if you get it right, and which one you used
Hi! Can someone get this answer? Solve the system if linear equations by substitution
Answers
Answer:
x=-4 y=5
Step-by-step explanation:
3×(16-4y)+4y=8
48-12y+4y=8
-8y=-40
y=5
x=16-4×5
x=-4
y = 4.99x
x = # of items
y = total cost
Answers
Answer:
here/
Step-by-step explanation:
charges a cost per pound plus a fixed fee to ship a package. the total cost, f(x) ... f(x)=4.99x+5.75. which part of the function represents the fixed fee? 2 ... the cost of items purchased and y represents the total cost of shipping.
2 answers
·
13 votes:
the fixed fee is the part of $5.75
pls answert this within 1 hour or else i will be DOOMED
Answers
In the fractions prompts given, the correct output are:
(1 ÷2) ÷ 4 = 1/8(1 ÷5) ÷ 2 = 1/10(1 ÷3) ÷ 5 = 1/15(1 ÷4) ÷ 4 = 1/16The solution to the problem for (1/2) ÷ 3 is given below.What is a fraction?A fraction represents a part of a whole. It consists of a numerator and a denominator, with the numerator indicating the number of parts and the denominator indicating the total number of parts.
The calculations are given as follows;
1 )
= (1/2) x (1/4) [dividing by a number is the same as multiplying by its reciprocal]
= 1/8
2) (1/5) ÷ 2
= (1/5) x (1/2)
= 1/10
3)
(1/3) ÷ 5
= (1/3) x (1/5)
= 1/15
4) (1/4) ÷ 4
= (1/4) x (1/4)
= 1/16
5) One day, Amy baked a cake and wanted to divide it equally among 3 of her friends. She realized she only had half a cake left, so she decided to divide it into equal parts. Each friend received 1/6 of a cake. To check her calculation, she multiplied 1/6 by 3 and got 1/2. Thus, (1/2) ÷ 3 = 1/6, since dividing by 3 is the same as multiplying by 1/3.
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