For any a, b belongs to Z, if ab is odd, then a² + b³ is even.
To prove that for any integers a and b, if ab is odd, then a² + b³ is even, we can use proof by contradiction.
Assume that there exist integers a and b such that ab is odd, but a² + b³ is not even (i.e., it is odd).
Since ab is odd, we can write it as ab = 2k + 1, where k is an integer.
Now, let's assume that a² + b³ is odd. This means that a² + b³ = 2m + 1, where m is an integer.
From the equation ab = 2k + 1, we can express a as a = (2k + 1) / b.
Substituting this into the equation a² + b³ = 2m + 1, we get ((2k + 1) / b)² + b³ = 2m + 1.
Expanding the equation, we have (4k² + 4k + 1) / b² + b³ = 2m + 1.
Multiplying both sides by b², we get 4k² + 4k + 1 + b⁵ = (2m + 1)b².
Rearranging the terms, we have 4k² + 4k + 1 = (2m + 1)b² - b³.
Notice that the left side (4k² + 4k + 1) is always odd because it is the sum of odd numbers.
The right side ((2m + 1)b² - b³) is also odd because it is the difference of an odd number and an odd number (odd - odd = even).
However, we have a contradiction since an odd number cannot be equal to an even number.
Therefore, our assumption that a² + b³ is odd must be false.
Consequently, if ab is odd, then a² + b³ must be even for any integers a and b.
Hence, the statement is proven.
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How many zeros appear at the end of 115!? Do not compute 115!.
Your argument must come from prime factorizations to receive
credit.
there will be 27 zeros at the end of 115!.
To determine the number of zeros at the end of 115!, we need to consider the prime factorization of the number and examine how many factors of 5 are present.
A zero at the end of a factorial occurs when there is a factor of 10 present, which is equivalent to having both factors of 2 and 5. Since the number of factors of 2 is usually abundant, the crucial factor is the number of factors of 5.
In the prime factorization of 115!, the factors of 5 arise from the multiples of 5 (5, 10, 15, 20, ...) as well as higher powers of 5 (25, 50, 75, ...). We need to determine how many multiples of 5, multiples of 25, multiples of 125, and so on are present.
1. Multiples of 5: The number of multiples of 5 in 115! is given by ⌊115/5⌋ = 23.
2. Multiples of 25: The number of multiples of 25 in 115! is given by ⌊115/25⌋ = 4.
3. Multiples of 125: The number of multiples of 125 in 115! is given by ⌊115/125⌋ = 0 since there are no numbers in the range 1 to 115 that are multiples of 125.
Adding up these counts, we have 23 + 4 = 27 factors of 5.
Therefore, there will be 27 zeros at the end of 115!.
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find vertex of this function
Answer:
Step-by-step explanation:
(-1,-9)
The length of a rectangle is two more than triple the width. If the perimeter is 166 inches, what are the dimensions of the rectangle?
The dimensions of the rectangle are length = 62.75 inches and width = 20.25 inches.
The given problem states that the length of a rectangle is two more than triple the width.
If the perimeter is 166 inches, what are the dimensions of the rectangle? Let's solve the problem,
Step 1
Given, The length of the rectangle = l
Width of the rectangle = w
The perimeter of the rectangle = 166 inches
The formula for the perimeter of a rectangle is,
Perimeter = 2(l + w)
So, 166 = 2(l + w)166/2 = l + w83 = l + w ----(1)
Step 2
According to the given problem, The length of a rectangle is two more than triple the width
Therefore,
l = 2 + 3w
Substitute this value in equation (1)
83 = (2 + 3w) + w
83 = 2 + 4w
83 - 2 = 4w
81 = 4w
w = 81/4
w = 20.25 (approx)
Step 3
We have width w = 20.25 inches.
We can find the length l by substituting w in l = 2 + 3w
So,
l = 2 + 3(20.25)
= 2 + 60.75
= 62.75
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the quotient of 17 and z
Answer:
17÷z = quotient
Step-by-step explanation:
quotient = ÷
What is the probability of spinning a number greater than 4?
Answer:
Please upload the full question.....here the total number of outcomes is not mentioned and hence it can't be solved.
Answer:
Hence, the required probability of getting a number greater than 4, P(E) = 1/3.
Step-by-step explanation:
At the state fair, admission at the gate is $9. In addition, the cost of each ride is $2. Suppose that Reuben will go on x rides.
Reuben wants the total number of dollars he spends on admission and rides to be fewer than . Using the values and variables given, write an inequality describing this.
Answer:
i think you forgot to add how little he wants to spend
Step-by-step explanation:
Reuben wants the total number of dollars he spends on admission and rides to be fewer than ??? whats the number that he wants to spend
I need this answer as soon as possible
The perimeter is the distance all the way around. So it's the sum of the lengths of all 4 sides.
From the picture, you can clearly see the lengths of all 4 sides.
Writum down and adum up !
Cheryl's making trail mix for a friend she always uses three cups of almonds for every four cups of cashews if Cheryl wants to make a larger batch of trail mix which of the following has the same ratio of almonds cashews
A. 9 cups of almonds for every 20 cups of cashews
B. 6 cups of almonds for every 12 cups of cashews
C. 9 cup of almonds for every 12 cups of cashews
D. 12 cups of almonds for every 20 cups of cashews
Answer:
c
Step-by-step explanation:
3x3=9
4x3=12
Answer:
C
Step-by-step explanation:
3 cups of almonds / 4 cups of cashews
So to get 9 cups of almonds, we need to multiply almonds and cashews by 3
3*3 = 9 cups of almonds
4*3 = 12 cups of cashews
So the correct answer is C
Use the binomial series to find a Taylor polynomial of degree 3 for 1 91 +32 T3(0) X + c? + 23
The Taylor polynomial of degree 3 for the function 1/(1-2x) centered at x=0 is (1+2x+4x²+8x³).
Explanation: Given, 1/(1-2x) = ∑n=0 to infinity of 2^n * x^n The above series is the binomial series for (1+x)^n where n=-1Using the binomial series for n=-1, we have1/(1-2x) = ∑n=0 to infinity of 2^n * x^n= ∑n=1 to infinity of 2^(n-1) * x^(n-1)= 1 + ∑n=1 to infinity of 2^n * x^nTaking up to degree 3, we get1/(1-2x) = 1 + 2x + 4x² + 8x³ + ...Therefore, the Taylor polynomial of degree 3 for 1/(1-2x) is 1 + 2x + 4x² + 8x³.
An infinite sum of words that are expressed in terms of a function's derivatives at a single point is known as the Taylor series or Taylor expansion of a function in mathematics. Near this point, the function and the sum of its Taylor series are equivalent for the majority of common functions. for Brook Taylor, who introduced the Taylor series in 1715, they are named for him. In honour of Colin Maclaurin, who made great use of this unique example of Taylor series in the middle of the 18th century, a Taylor series is sometimes known as a Maclaurin series where 0 is the point at which the derivatives are taken into account.
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Suppose f(x,y,z) = In(x + 2y2 + 3z"). Find the following partial derivatives. a. fx b. fz c.d2f/dzdx.
The partial derivatives are as follows :
(a) fx = 1 / (x + 2y^2 + 3z^3)
(b) fz = 3z^2 / (x + 2y^2 + 3z^3)
(c) d^2f/dzdx = -3z^2 / (x + 2y^2 + 3z^3)^2
To find the partial derivatives of the function f(x, y, z) = ln(x + 2y^2 + 3z^3), we differentiate with respect to each variable while treating the other variables as constants.
(a) Partial derivative with respect to x (fx):
To find fx, we differentiate the function f(x, y, z) with respect to x while treating y and z as constants. The derivative of ln(u) with respect to u is 1/u, so we have:
fx = d/dx ln(x + 2y^2 + 3z^3) = 1 / (x + 2y^2 + 3z^3)
(b) Partial derivative with respect to z (fz):
To find fz, we differentiate the function f(x, y, z) with respect to z while treating x and y as constants. Again, applying the derivative of ln(u), we get:
fz = d/dz ln(x + 2y^2 + 3z^3) = 3z^2 / (x + 2y^2 + 3z^3)
(c) Second partial derivative with respect to z and x (d^2f/dzdx):
To find d^2f/dzdx, we differentiate fz with respect to x while treating y and z as constants. We differentiate fx with respect to z while treating x and y as constants, and then take the derivative of the result with respect to z. It can be written as:
d^2f/dzdx = d/dx (d/dz ln(x + 2y^2 + 3z^3)) = d/dx (3z^2 / (x + 2y^2 + 3z^3))
= -3z^2 / (x + 2y^2 + 3z^3)^2
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Proving a parallelogram side theorem.
Given ABCD is a parallelogram.
Prove: AB≈CD and BC ≈ DA
Answer:
just did it on edg, 2021
Hence, AB≈CD and BC ≈ DA
What is a parallelogram?
A four sided closed figure with all its sides parallel to its opposite side.
Consider two triangles ΔABD and ΔBCD
∠BCD=∠BCA( opposite angles are always equal)
∠ABD=∠BDC (AD||BC)
∠ADB=∠DBC(AD||BC)
ΔABD ≅ ΔBCD
AB≈CD by CPCTC
BC ≈ DA by CPCTC
Hence, proved
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There are 180 trees in gardner grove orchard, and 18 of them are pears. What percent of the trees are pear trees?
Q1
The sum of the first 68 positive odd integers is ?
Q2
The degree of recurrence relation an = 2an-2 + 5an-49 is ??
Q3
In how many ways can an organization containing 19 members elect a president, treasurer and secretary (assuming no person is elected to more than one position)?
Q4
The Greatest Common Divisor (GCD) of 28 × 37 × 58 and 23 × 33 × 54 is ??
Q1: Sum of first 68 positive odd integers.
Let's represent the first 68 positive odd integers by: 1, 3, 5, 7, ..., 135, 137. The first term, a = 1. The last term, l = 137And, the number of terms, n = 68We need to find the sum of these terms. To find the sum of an arithmetic series, we use the following formula: Sn = n/2[2a + (n-1)d]. Here, d = common difference. Since the given sequence is of odd numbers, the difference between any two consecutive terms is 2. So, d = 2. Put these values in the formula to get: Sn = 68/2[2(1) + (68-1)2], Sn = 34[2 + 135], Sn = 68 × 67Sum of first 68 positive odd integers = 4546.
Q2: Degree of recurrence relation. To find the degree of a recurrence relation, we find the largest value of n in the relation. Here, an = 2an-2 + 5an-49The largest value of n in the relation is n = 49. So, the degree of the recurrence relation is 49.
Q3: Number of ways to elect office bearers in an organization. Let's assume that the 19 members of the organization are named M1, M2, M3, ..., M19. The president can be elected in 19 ways. After the president is elected, the treasurer can be elected in 18 ways. After the treasurer is elected, the secretary can be elected in 17 ways. Therefore, the total number of ways in which the president, treasurer, and secretary can be elected is:19 × 18 × 17 = 5,814.
Q4: Greatest Common Divisor (GCD)To find the GCD of two numbers, we need to find their prime factors.28 × 37 × 58 = 2² × 7 × 37 × 2 × 29 = 2³ × 7 × 29 × 37Similarly, 23 × 33 × 54 = 23 × 3² × 2 × 3 × 3 × 2 × 3 = 2³ × 3⁵ × 23.
The common prime factors are 2³ and 23. So, the GCD is: 2³ × 23 = 184. The GCD is 184.
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Joey made strawberry jam and raspberry jam. He made enough strawberry jam to fill 1/2 of a jar. If he made 2/5 as much raspberry jam as strawberry jam, how many jars will the raspberry jam fill?
Answer: 1/5 of a jar
Step-by-step explanation:
1/2 times 2/5= .2 = 1/5.
Answer:
1/5 of a jar
Step-by-step explanation:
8. (08.02 lc)complete the square to transform the expression x2 6x 5 into the form a(x − h)2 k. (1 point)(x 6)2 4(x 6)2 − 4(x 3)2 − 4(x 3)2 4
The expression [tex]x^{2}[/tex] + 6x + 5 can be completed by transforming it into the form a(x - h)^2 + k.
To complete the square, we want to rewrite the quadratic expression x^2 + 6x + 5 in a perfect square trinomial form. We can achieve this by adding and subtracting a constant term inside the parentheses.
Starting with the given expression: x^2 + 6x + 5
To complete the square, we need to take half of the coefficient of x and square it. Half of 6 is 3, and squaring 3 gives us 9. So, we add and subtract 9 inside the parentheses:
x^2 + 6x + 5 = (x^2 + 6x + 9 - 9) + 5
Now, we can group the first three terms as a perfect square trinomial and simplify:
(x^2 + 6x + 9 - 9) + 5 = (x + 3)^2 - 9 + 5
Simplifying further, we have:
(x + 3)^2 - 4
Therefore, the expression x^2 + 6x + 5 can be written in the form a(x - h)^2 + k as (x + 3)^2 - 4.
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solve the given differential equation. y(ln(x) − ln(y)) dx = (x ln(x) − x ln(y) − y) dy
To solve the given differential equation:
y(ln(x) - ln(y)) dx = (x ln(x) - x ln(y) - y) dy
We can start by rearranging the terms:
y ln(x) dx - y ln(y) dx = x ln(x) dy - x ln(y) dy - y dy
Next, we can integrate both sides of the equation:
∫ y ln(x) dx - ∫ y ln(y) dx = ∫ x ln(x) dy - ∫ x ln(y) dy - ∫ y dy
To integrate the left-hand side, we can use integration by parts. Let's denote u = ln(x) and dv = y dx. Then, du = (1/x) dx and v = xy. Applying integration by parts, we have:
∫ y ln(x) dx = xy ln(x) - ∫ (1/x)(xy) dx
= xy ln(x) - ∫ y dx
= xy ln(x) - yx + C1
where C1 is the constant of integration.
Similarly, integrating the other terms:
∫ y ln(y) dx = xy ln(y) - yx + C2
∫ x ln(x) dy = (x^2 ln(x))/2 - ∫ (x^2)(1/x) dy
= (x^2 ln(x))/2 - ∫ x dy
= (x^2 ln(x))/2 - (x^2)/2 + C3
∫ x ln(y) dy = (x^2 ln(y))/2 - ∫ (x^2)(1/y) dy
= (x^2 ln(y))/2 - ∫ (x^2/y) dy
= (x^2 ln(y))/2 - x^2 ln(y) + ∫ x dy
= (x^2 ln(y))/2 - x^2 ln(y) + (x^2)/2 + C4
∫ y dy = (y^2)/2 + C5
Substituting these results back into the original equation:
xy ln(x) - yx + C1 - xy ln(y) + yx - C2 = (x^2 ln(x))/2 - (x^2)/2 + C3 - (x^2 ln(y))/2 + x^2 ln(y) - (x^2)/2 + C4 - (y^2)/2 - C5
Simplifying:
xy ln(x) - xy ln(y) = (x^2 ln(x))/2 - (x^2 ln(y))/2 - (y^2)/2 + C
where C = C1 - C2 + C3 + C4 - C5.
We can further simplify this equation:
xy (ln(x) - ln(y)) = (x^2 ln(x) - x^2 ln(y) - y^2)/2 + C
Finally, dividing both sides by (ln(x) - ln(y)), we get:
xy = (x^2 ln(x) - x^2 ln(y) - y^2)/(2(ln(x) - ln(y))) + C
This is the general solution to the given differential equation.
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Find the value of x. I WILL MARK YOU BRAINLIEST!!!
Answer:
4x+3x+2x=180
9x=180
x=20
Which one of these is in scientific notation?
(47 points)
Answer:
8.98 * 10^6 is the scientific notation
Step-by-step explanation:
The table below gives the distribution of milk
chocolate M&M's
Color
NA
DO
Probability
0.13
0.13
0.14
0.16
0.20
0.24
If a candy is drawn at random, what is the probability
that it is not orange or red?
Answer: .67
Step-by-step explanation:
got it right on acellus
A 28 ft tall house casts a shadow that is 35 ft long. A umbrella outside the house casts a shadow that is 16 7/8 long. How tall is the umbrella?
Answer:
The unbrella is 13.5 feet tall.
Step-by-step explanation:
35 / 28 = 1.25
1.25x = 16 7/8
x = (16 7/8) / 1.25
x = 13.5 ft
Which graph shows a function where f(2)=4?
Answer:
A
Step-by-step explanation:
Answer:
the one that is a horizontal straight line on x axis that lies on 2
AHH HELP ME PLS
Choose the values of x that are solutions to the inequality 5 <,
Select all that apply
Ax=2
= 6
E *=-7
F1 = 10
Answer:
x = 5
x = 6
x = 10
Step-by-step explanation:
NOTE: IT HAS TO BE MORE THAN 5 OR EQUAL TO 5
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What’s the answer to this?!!!
Answer:
77 meters
Step-by-step explanation:
For a right triangle, the formula for side length is
a^2 + b^2 = c^2 where c is the hypotenuse (opposite the right angle)
36^2 + b^2 = 85^2
1296 + b^2 = 7225
b^2 = 5,929
find the square root of both sides:
b = 77
Please let me know if you have questions.
Which of the following is the first step in the decision-making process?
a. Analyze the problem and its causes
b. Generate alternatives
c. Solicit and analyze feedback
d. Identify the problem
The correct answer is D to identify the problem.
What is the decision-making process?
The decision-making process is the method by which a judgment or a decision is reached. It involves identifying the problem or decision to be made, analyzing potential courses of action, evaluating alternatives, and selecting the best possible solution.
It's a structured process that assists in making effective decisions, and it can be useful in both personal and professional contexts. What is the first step in the decision-making process?
The first step in the decision-making process is to identify the problem. This entails defining the issue that requires a decision to be made. It's a crucial step because without accurately identifying the issue or problem, it's impossible to make the best decision.
Analyzing the problem and its causes (A), generating alternatives (B), and soliciting and analyzing feedback (C) are all critical components of the decision-making process, but they come after the problem has been identified.
As a result, option D, identifying the problem, is the first step in the decision-making process.
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A part manufactured at a factory is known to be 12.05 cm long on average, with a standard deviation of 0.350. One day you suspect that that the part is coming out a little longer than usual, but with the same deviation. You sample 14 at random and find an average length of 12.20. What is the z-score which would be used to test the hypothesis that the part is coming out longer than usual?
The z-score to test the hypothesis that the part is coming out longer than usual is approximately 1.61.
Sample mean = x = 12.20 cm
Population mean = μ = 12.05 cm
Standard deviation = σ = 0.350 cm
Sample size = n = 14
A hypothesis is an informed prediction regarding the solution to a scientific topic that is supported by sound reasoning. there is the expected result of the experimentation even if there is not proved in an experiment.
Calculating the z-score -
[tex]z = (x - u) / (\alpha / \sqrt n)[/tex]
Substituting the values -
[tex]z = (12.20 - 12.05) / (0.350 / \sqrt{14)[/tex]
= z = 0.15 / (0.350 / √14)
= 0.093
Substituting the value again into the formula:
z = 0.15 / 0.093
= 1.61
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a 18 The number 21 has no composite factors. What is another number that has no composite factors? A. 27- B. 52- C. 77- D. 81-
The number that has no composite factors is (c) 77
How to determine another number that has no composite factorsFrom the question, we have the following parameters that can be used in our computation:
Number = 21
The factors of 21 are 3 and 7
These factors are composite numbers because they are prime numbers
using the above as a guide, we have the following:
The number 77 has no composite factors
This is so because
77 = 7 * 11
These factors are composite numbers because they are prime numbers
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Find the first three terms of x[n] using power series expansion if X(z) 2z3 + 13z2 + 7 73 + 722 + 2z + 1 =
The first three terms of x[n] using the power series expansion are x[0] = 73, x[1] = 2, and x[2] = 13.
We can select the first three terms of x[n] using the power series development by expressing the given articulation X(z) as a polynomial in z. We should modify the articulation as follows to obtain the power series development: By comparing the given expression to the power series form, the coefficients can be identified: X(z) equals 2z3, 13z2, 7z, 73, 722/z, 2/z, and 1: a0 rises to 73, a1 approaches 2, a2 approaches 13, and a3 approaches 7. X(z) = a0, a1z, a2z2, a3*z3, and... Consequently, the following are the first three terms of x[n]:
The initial three terms of x[n] are provided by the power series development: x[0] = a0; x[1] = a1; x[2] = a2; x[0] = a0; The values of x[0] and x[1] are 73, 2, and 13.
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Work out the circumference of this circle.
Take a to be 3.142 and write down all the digits given by your calculator.
16.8 cm
Answer: 52.7856cm
Step-by-step explanation:
Circumference of a circle = 2πr
Note that radius = Diameter / 2 = 16.8/2 = 8.4cm
Circumference = 2πr
= 2 × 3.142 × 8.4
= 52.7856cm
Therefore, the circumference of the circle is 52.7856cm
Find the eighth term of a geometric sequence for which a,3 = 35 and r= 7.
Answer:
588245.
Step-by-step explanation:
nth term = an = a1 r^(n-1) where a1 = the first term
a3 = 35 = a1 7^(3 - 1)
35 = a1* 49
a1 = 35/49 = 5/7
So the 8th term = (5/7)* (7)^7
= 588245
Renee is a sales associate at a store. She earns $80 a week plus a 15% commission on her sales. Last week, she sold $200 worth of items. What is the total amount Renee earned for the week? How much did she earn from commission?
Renee earned a total of $
for the week. The amount she earned from commission was $
.
Answer:
The amount earned for the week=$110
Amount earned from commission =$30
Step-by-step explanation:
commission earned on sales = $200×15%= $30
total amount for the week=$80 +$30= $110