For Q5, Q6 use a direct proof, proof by contraposition or proof by contradiction. 5) Prove that for every n e Z, n² - 2 is not divisible by 4.
Answers
To prove that for every integer n, n² - 2 is not divisible by 4, a direct proof will be used. To prove the statement, we will employ a direct proof, showing that for any arbitrary integer n, n² - 2 cannot be divisible by 4.
Assume that n is an arbitrary integer. We will consider two cases: when n is even and when n is odd.
Case 1: n is even (n = 2k, where k is an integer)
In this case, n² is also even since the square of an even number is even. Therefore, n² - 2 = 2m, where m is an integer. However, 2m is divisible by 2 but not by 4, so n² - 2 is not divisible by 4.
Case 2: n is odd (n = 2k + 1, where k is an integer)
In this case, n² is odd since the square of an odd number is odd. Therefore, n² - 2 = 2m + 1 - 2 = 2m - 1, where m is an integer. 2m - 1 is not divisible by 4 as it leaves a remainder of either 1 or 3 when divided by 4.
In both cases, we have shown that n² - 2 is not divisible by 4. Since these cases cover all possible integers, the statement holds true for all values of n.
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To prove that for every integer n, n² - 2 is not divisible by 4, a direct proof will be used. To prove the statement, we will employ a direct proof, showing that for any arbitrary integer n, n² - 2 cannot be divisible by 4.
Assume that n is an arbitrary integer. We will consider two cases: when n is even and when n is odd.
Case 1: n is even (n = 2k, where k is an integer)
In this case, n² is also even since the square of an even number is even. Therefore, n² - 2 = 2m, where m is an integer. However, 2m is divisible by 2 but not by 4, so n² - 2 is not divisible by 4.
Case 2: n is odd (n = 2k + 1, where k is an integer)
In this case, n² is odd since the square of an odd number is odd. Therefore, n² - 2 = 2m + 1 - 2 = 2m - 1, where m is an integer. 2m - 1 is not divisible by 4 as it leaves a remainder of either 1 or 3 when divided by 4.
In both cases, we have shown that n² - 2 is not divisible by 4. Since these cases cover all possible integers, the statement holds true for all values of n.
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Related Questions
domain and range of piecewise functions calculator
Answers
Answer:
yes
Step-by-step explanation:
Can I get a good explanation on SAS Triangle Congruence? (35 points)
Answers
\(\huge{ \mathfrak{ \underline{ Answer} \: \: ✓ }}\)
SAS is a congruency criteria which is used to prove that two triangles are congruent, if two corresponding sides and an angle between them is equal in both the triangles.
For example - In the attached picture,
KL = XY∠KLM = ∠XYZLM = YZTherefore, ΔKLM ≅ ΔXYZ
(By SAS criteria)
\(\mathrm{ ☠ \: TeeNForeveR \:☠ }\)
Which equation, together with y = -1. 5x + 3, makes a system with one solution?
Y= -1. 5x + 6
Y= -1. 5x
2Y= -3x + 6
2Y + 3x = 6
Y= -2x + 3
Answers
The equations that, together with y = -1.5x + 3, make a system with one solution are: 1. Y = -1.5x + 6, 4. 2Y + 3x = 6, 5. Y = -2x + 3
To find the equation that makes a system with one solution when paired with y = -1.5x + 3, we need to find an equation with a different slope and/or a different y-intercept.
Let's analyze each given equation:
1. Y = -1.5x + 6: This equation has the same slope as y = -1.5x + 3 but a different y-intercept, so it has one solution.
2. Y = -1.5x: This equation has the same slope as y = -1.5x + 3, and therefore has an infinite number of solutions (or is dependent), so it's not the answer.
3. 2Y = -3x + 6: To compare with the given equation, let's put it in the same format. Divide both sides by 2 to get Y = -1.5x + 3, which is the same as the given equation, so it's not the answer.
4. 2Y + 3x = 6: To compare with the given equation, let's put it in the same format. Divide both sides by 2 to get Y + 1.5x = 3. This equation has a different slope, so it has one solution.
5. Y = -2x + 3: This equation has a different slope and y-intercept, so it has one solution.
Thus, the equations that, together with y = -1.5x + 3, make a system with one solution are:
1. Y = -1.5x + 6
4. 2Y + 3x = 6
5. Y = -2x + 3
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26) Anjali and Rob are selling cookie dough for a school fundraiser. Customers can buy packages of
sugar cookie dough and packages of oatmeal cookie dough. Anjali sold 6 packages of sugar cookie
dough and 2 packages of oatmeal cookie dough for a total of $116. Rob sold 9 packages of sugar
cookie dough and 12 packages of oatmeal cookie dough for a total of $318. Find the cost each of
one package of sugar cookie dough and one package of oatmeal cookie dough.
Answers
Answer:
Sugar cookie dough packages cost 14 dollars each, and oatmeal cookie dough packages cost 16 dollars each.
Step-by-step explanation:
Let s equal the cost of a sugar cookie dough package, and o equal the cost of oatmeal cookie dough packages.
6s + 2o = 116
9s+12o = 318
therefore,
18s+6o = 348
18s+24o = 636
Subtracting them, we get -18o = -288, or 18o = 288
Therefore, o = 16, and 6s+32=116, so 6s=84, so s = 14
Sarah exchanged 1,000 US dollars for the Croatian Kuna. The exchange rate was 6.72 Kuna for $1. How many Croatian Kuna did Sarah get?
Answers
Answer:
6720 Kuna
Step-by-step explanation:
multiply the rate for one dollar by the amount of money your are converting
6.72 x 1000 = 6720
A pyramid has a square base of length of 12m. Given the slant height of the pyramid is 15 m, draw its net and hence, find its total surface area.
Answers
Answer:
the surface area is 180 if u use multiplication
Step-by-step explanation:
the surface is 0.8 if you use devide
the surface is 3 if u use subtract
and the surface is 27 if you use addition
correct me if I'm wrong :)
Gabriel needs d more dollars to buy a new video game console that costs $156.75. He has saved $98.26. Solve the equation $98.26+d=$156.75to find how much more money Gabriel needs.
Answers
Answer:
d=58.5 hope this helps
Step-by-step explanation:
A triangle has vertices at B(-3,0), C(2, -1), D(-1, 2). Which transformation would produce an image with vertices B"(-2, 1), C"(3, 2), D"(0, -1)? (x,y) → (X, -y). (x,y) → (X + 1, y + 1) (x,y) → (-x, y). (x,y) → (X + 1, y + 1) (x,y) → (X. -Y). (x,y) → (X + 2, Y + 2) (x,y) → (-x, y). (x,y) → (X + 2, Y + 2)
Answers
The transformation that would produce an image with vertices B"(-2, 1), C"(3, 2), D"(0, -1) from the original vertices B(-3,0), C(2, -1), D(-1, 2) is the transformation (x,y) → (X + 1, y + 1).
By applying the transformation (x,y) → (X + 1, y + 1), each point's x-coordinate is shifted one unit to the right (X = x + 1), and each point's y-coordinate is shifted one unit upward (Y = y + 1). This results in the image with the given coordinates B"(-2, 1), C"(3, 2), D"(0, -1).
The other transformations listed do not match the given image coordinates and would not produce the desired result.
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Answer:
(x, y) → (x, −y) → (x + 1, y + 1)
Step-by-step explanation:
Reflect the vertices over the x-axis by applying the transformation (x, y) → (x, -y):
B'(-3, 0) → B'(-3, 0)
C(2, -1) → C(2, 1)
D(-1, 2) → D(-1, -2)
Translate the reflected vertices by (1, 1):
B'(-3, 0) → B″(-3 + 1, 0 + 1) → B″(-2, 1)
C(2, 1) → C″(2 + 1, 1 + 1) → C″(3, 2)
D(-1, -2) → D″(-1 + 1, -2 + 1) → D″(0, -1)
So, the correct sequence of transformations is:
(x, y) → (x, -y) → (x + 1, y + 1)
(Image provided for more proof)
This is due tonight.. <3 THANKS!! im giving brainliest
Suppose Chau has 8.04 pounds of seed. If it takes 6.7 pounds of seed to plant one acre of grass, how many acres of grass can be planted?
Answers
Answer:
its simple ans is 1.2 acres
Determine the Sum of the first 20 terms in the sequence: 3 + 7 + 11 + 15 ....
Answers
Answer:
820
Step-by-step explanation:
whats the gamble formula
Answers
Gambling formula
is the fraction of the current bankroll to wager. is the probability of a win. is the probability of a loss ( ). is the proportion of the bet gained with a win. E.g. If betting $10 on a 2-to-1 odds bet, (upon win you are returned $30, winning you $20), then .
hope it helps..!!!
5y-3x+6x-3y=3x+____y
Answers
Answer: 5 x = 9 i think i could be wrong
Step-by-step explanation:
Answer:
5y - 3x + 6x - 3y = 3x + 2y
Step-by-step explanation:
To fill in the missing term in the equation:
5y - 3x + 6x - 3y = 3x + ____y
We can simplify the equation by combining like terms:
(5y - 3y) + (6x - 3x) = 3x + ____y
2y + 3x = 3x + ____y
To make the equation balanced, the missing term should be 2y.
Therefore, the filled equation is:
5y - 3x + 6x - 3y = 3x + 2y
plz help asap. i beg
Answers
The answer is -15.
Because others are rational numbers but not integers.
-15 is the only rational number and integer.
Given z=f(x
1
,x
2
)=6x
1
2
+12x
2
2
with constraint c=g(x
1
,x
2
)⇒90=x
1
+x
2
, solve for the optimal values for the Lagrangian objective function. Finally, verify whether your optima maximized or minimized the Lagrange.
Answers
The optima of the Lagrangian objective function is minimized at z = 54,000. Also the Lagrange multiplier solution corresponds to a minimum.
The optimal values for the Lagrangian objective function can be determined by solving the given optimization problem using Lagrange multipliers. We have the objective function z = 6x₁² + 12x₂² and the constraint g(x₁, x₂) = 90 = x₁ + x₂.
To find the optimal values, we form the Lagrangian function L(x₁, x₂, λ) = f(x₁, x₂) - λ(g(x₁, x₂) - 90). Here, λ is the Lagrange multiplier.
Taking the partial derivatives with respect to x₁, x₂, and λ, and setting them to zero, we obtain the following equations:
∂L/∂x₁ = 12x₁ - λ = 0
∂L/∂x₂ = 24x₂ - λ = 0
∂L/∂λ = x₁ + x₂ - 90 = 0
Solving these equations simultaneously, we find x₁ = 30, x₂ = 60, and λ = 360. Substituting these values back into the objective function, we get z = 6(30)² + 12(60)² = 54,000.
To determine whether this is a maximum or minimum, we can examine the second partial derivatives of the Lagrangian. Calculating the second partial derivatives, we have:
∂²L/∂x₁² = 12
∂²L/∂x₂² = 24
Since both second partial derivatives are positive, we can conclude that the Lagrange multiplier solution corresponds to a minimum. Therefore, the optima of the Lagrangian objective function is minimized at z = 54,000.
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A pairs of fair dice is tossed . What is the probability of not getting a sum 5 or 9 ?
Answers
Answer:
28/36 or 7/9
Step-by-step explanation:
there are 36 possible answers
4 of them equal 5 (1+4, 2+3, 3+2, 4+1)
4 of them equal 9 (3+6, 4+5, 5+4, 6+3)
8-36 = 28
an explanation of an event that is based on repeated observations and experiments is a
Answers
An explanation of an event that is based on repeated observations and experiments is a scientific theory
A thorough, well-researched explanation of a natural occurrence constitutes a scientific theory.
We frequently use the word "theory" to refer to a hypothesis or informed guess in common speech, but in science, a theory is an explanation that is the result of a substantial and repeated investigation. The purpose of theories is to explain a broad concept; they are not intended to establish facts
Some of the most well-known and complicated events are explained by scientific theories. The theories of relativity, evolution, and gravity are a handful of the most well-known scientific concepts.
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Suppose that you plan on taking a summer job selling magazine subscriptions. The magazine company will pay you $ 1.25 for selling the first subscription. For each additional subscription sold, the magazine company will pay you 15 cents more than what was paid for the previous subscription. How much will you earn by selling 175 magazine subscriptions?
Answers
Answer: $27.35
Step-by-step explanation:
First let's find the variables.We will use x to represent the number of subscription sold and y we will represent the amount of money you will get paid.
So we know that , if you sell One You will get $1.25 so we could represent it by the point (1,1.25) .
Now the second statement says that if you sell an additional subscription you will get an additional 15 cents or $0.15.Which means if you sell the second subscription, you will get $1.40 because it is an additional 15 cents. We can also represent that by the point (2,1.40)
Now using the two points come up with an equation.
(1,1.25)
(2,1.40) Find the slope and y intercept.
TO find the slope find the difference between y and divide it by the difference in x.
1.25 - 1.40 = 0.15
1 - 2 = 1
0.15/ 1 = 0.15 So the slope is 0.15
Now use the point to find the y intercept. using the y-intercept formula,
y = mx + b
1.25 = 0.15(1) + b
1.25 = 0.15 + b
-0.15 0.15
b = 1.1
We can now write the equation y = 0.15x + 1.1 to represent the whole situation where x is still the number of subscriptions sold and y is the money you will get paid. Now to find how much you will earn for 175 subscriptions, you will input 175 for x and solve for y.
y = 0.15(175) + 1.1
y = 26.25 + 1.1
y = 27.35
So you will earn $27.35 for 175 subscriptions
do blood-building drugs help brain development in babies born prematurely? researchers randomly assigned 53 babies, born more than a month premature and weighing less than 3 pounds, to one of three groups. babies either received injections of erythropoietin (epo) three times a week, darbepoetin once a week for several weeks, or no treatment. results? babies who got the medicines scored much better by age 4 on measures of intelligence, language, and memory than the babies who received no treatment. what is the purpose of randomly assigning the babies to the three treatments? random assignment eliminates any bias in the experiment. random assignment helps to avoid confounding and reduces variation in responses. random assignment helps to create three groups of infants that are roughly equivalent at the beginning of the study. random assignment helps us distinguish the effects of the treatments form chance difference between the groups. random assignment allows us to show how different treatments cause different effects.
Answers
The correct option regarding the random assignment of babies in the treatment is given as follows:
Random assignment helps us distinguish the effects of the treatments form chance difference between the groups.
Randon AssignmentRandom assignment is used for control in experimental researches, to strengthen the validity of an experiment, distinguish the effects of the treatments form chance difference between the groups.
In these experiments, the effect of an independent variable is studied in one or multiple dependent variables in the study, and then the random assignment increases the validity of these effects.
Thus the correct option is given as follows:
Random assignment helps us distinguish the effects of the treatments form chance difference between the groups.
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The radioactive isotope 226Ra has a half-life of approximately 1599 years. There are 80g of 226Ra now.
(1) How much of it remains after 1,700 years? (Round your answer to three decimal places.)
(2) How much of it remains after 17,000 years? (Round your answer to three decimal places.)
Answers
1) 0.080 grams of 226Ra would remain after 17,000 years.
2) 44.000 grams of 226Ra would remain after 1,700 years.
To calculate the remaining amount of a radioactive isotope after a certain time, we can use the formula:
\(N(t) = N₀ * (1/2)^{(t / T_{ \frac{1}{2}} )}\)
Where:
N(t) is the remaining amount of the isotope after time t.
N₀ is the initial amount of the isotope
\(T_{ \frac{1}{2} }\) is the half-life of the isotope
Let's calculate the remaining amount of 226Ra for the given time periods:
(1) After 1,700 years:
\(N(t) = 80g * (1/2)^(1700 / 1599) \\ N(t) = 80g *(1/2)^(1.063165727329581) \\ N(t) ≈ 80g * 0.550 \\ N(t) ≈ 44.000g\)
(rounded to three decimal places)
Therefore, approximately 44.000 grams of 226Ra would remain after 1,700 years.
(2) After 17,000 years:
\(N(t) = 80g * (1/2)^(17000 / 1599) \\ N(t) = 80g * (1/2)^(10.638857911194497) \\ N(t) ≈ 80g * 0.001 \\ N(t) ≈ 0.080g \)
(rounded to three decimal places)
Therefore, approximately 0.080 grams of 226Ra would remain after 17,000 years.
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Please help me step by step how to solve this quadratic equation 2a^2=-6+8a
Answers
The quadratic equation 2a^2 = -6 + 8a has two solutions: a = 3 and a = 1.
To solve the quadratic equation 2a^2 = -6 + 8a, we need to rearrange it into standard quadratic form, which is ax^2 + bx + c = 0, where a, b, and c are coefficients.
Step 1: Move all the terms to one side of the equation to set it equal to zero:
2a^2 - 8a + 6 = 0
Step 2: The equation is now in standard quadratic form, so we can apply the quadratic formula to find the solutions for 'a':
a = (-b ± √(b^2 - 4ac))/(2a)
Comparing with our equation, we have:
a = (-(-8) ± √((-8)^2 - 4(2)(6)))/(2(2))
Simplifying further:
a = (8 ± √(64 - 48))/(4)
a = (8 ± √16)/(4)
a = (8 ± 4)/(4)
Now, we can calculate the two possible solutions:
a1 = (8 + 4)/(4) = 12/4 = 3
a2 = (8 - 4)/(4) = 4/4 = 1
Therefore, the quadratic equation 2a^2 = -6 + 8a has two solutions: a = 3 and a = 1.
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Pls help meeeee thank u
Answers
Answer:
49
Step-by-step explanation:
A= a^2 = 7^2 =49
7333.33 to the NEAREST TENTH OF A CUBIC METER *will give brainliest* in a quiz hurry pls
Answers
Answer:
I think its 7330
which graph shows the solution
Answers
Answer:
Step-by-step explanation:
Solve:
-6n+5<11
-6n<6 (carry over the 5 by subtracting)
n>-1 (put "n" alone divide by -6, because dividing by a negative flip the symbol)
**Don't forget to carry the Negative Symbol on -6n
So, "A" is the answer
Test:
put in point
0>-1 ✔
or ....
-6n+5<11
-6(0)+5<11
5<11✔
So we know the line arrow goes in the positive direction
A biologist tags 25 birds with a band on their foot and lets them go back into the wild. The next year 100 total birds are captured and 5 of them have the foot bands. Enter an estimate of the total bird population. 20
Answers
Answer:
We can estimate a population of 500 birds.
Step-by-step explanation:
We can estimate the total bird population using a rule of three with the information given:
If we find 5 birds with the foot band among 100 birds captured, how many birds there are in total if inicially we put foot bands in 25 birds?
5 birds with foot band -> 100 birds captured
25 birds with foot band -> X total birds
X * 5 = 25 * 100
X = 25 * 100 / 5 = 500 birds
We can estimate a population of 500 birds.
Solve each inequality. p + 5 < 10 A. p < –15 B. p < 5 C. p < 15 D. p < –5
Answers
Answer:
B) p<5
Step-by-step explanation:
What do linear pairs make
Answers
= O EQUATIONS AND INEQUALITIES Solving a decimal word problem using a linear equation of the... Jayden Esp Leila received a $90 gift card for a coffee store. She used it in buying some coffee that cost $7.26 per pound. After buying the coffee, she had $60.96 left on her card. How many pounds of coffee did she buy?
Answers
Answer:
Jayden buys 4 pounds of coffee
Step-by-step explanation:
Let's say that p represents the number of pounds of coffee Jayden bought. She starts with $90, and for each pound she buys, she loses $7.26 dollars. Therefore, as she loses 7.26 dollars, we must subtract the amount she pays from $90. Then, as she loses 7.26 dollars for each pound, we multiply 7.26 by the number of pounds to find the money she spends.
We can thus write our equation as
90 - 7.26*p = final amount = 60.96
90-7.26p = 60.96
subtract 90 from both sides to isolate the p and its coefficient
-7.26p = -29.04
divide both sides by -7.26 to isolate the p
p = 4
Jayden buys 4 pounds of coffee
Christopher opened a savings account with $1800 that pays no interest. He deposits an additional $90 each week thereafter. How much money would Christopher have in the account 42 weeks after opening the account?
Answers
Answer:
I think 5580
Step-by-step explanation:
You move the 1800 away for a bit so since it's 42 weeks you have to multiply 90 and 42 then add that to 1800.
Please help with this question.
Answers
Answer:
The value of "h" = 32 unit
Step-by-step explanation:
Given:
In given parallelogram
Height for base 36 = 24
Height for base 27 = h
Find:
The value of "h"
Computation:
Area of parallelogram = Base x Height
For base 36
Area of parallelogram = 36 x 24
Area of parallelogram = 864 unit²
For base 27
Area of parallelogram = 27 x h
864 = 27 x h
The value of "h" = 864 / 27
The value of "h" = 32 unit
21/7 is rational and natural number only true or false
Answers
Answer:
True
Step-by-step explanation:
21/7=3
3 is a natural and rational number since it's a finite number with no repeating decimals and it's a whole number.