Answer:
Proof below
Step-by-step explanation:
General outlineLemma regarding remainders of non-small primes divided by 6Check [tex]p=2[/tex] and [tex]p=3[/tex] directlyProof the rest by contradictionLemmaIf [tex]p[/tex] is prime such that [tex]p \neq2[/tex] and [tex]p \neq 3[/tex], then [tex]p[/tex] divided by 6 has remainder of 1 or 5.
Case 1: p/6 has remainder 0, 2, or 4.
Then there exists some natural number [tex]n[/tex] such that either [tex]p=6n[/tex], [tex]p=6n+2[/tex], or [tex]p=6n+4[/tex]. However,
if [tex]p=6n[/tex], then [tex]p=2*3n[/tex],if [tex]p=6n+2[/tex], then [tex]p=2*(3n+1)[/tex], and if [tex]p=6n+4[/tex], then [tex]p=2*(3n+2)[/tex].By definition of divisibility, [tex]p[/tex] is divisible by 2. Since [tex]p \neq2[/tex], [tex]p[/tex] cannot be prime. This contradiction implies [tex]p[/tex] cannot have a remainder of 0, 2, or 4 when divided by 6.
Case 2: p/6 has remainder 3.
Then there exists some natural number [tex]n[/tex] such that [tex]p=6n+3[/tex]. However, if [tex]p=6n+3[/tex], then [tex]p=3*(2n+1)[/tex], so [tex]p[/tex] is divisible by 3. Since [tex]p \neq 3[/tex], [tex]p[/tex] is not prime. This contradiction implies [tex]p[/tex] cannot have a remainder of 3 when divided by 6.
Therefore, if [tex]p[/tex] is a prime number such that [tex]p \neq2[/tex] and [tex]p \neq 3[/tex], then when divided by 6, [tex]p[/tex] must have a remainder of 1 or 5.
Main proof of no prime trio with common difference of 70.Check p=2
Consider [tex]p=2[/tex]. Then [tex]q=2+70=72[/tex]. However, [tex]72=2*36[/tex], so [tex]q[/tex] is not prime. Hence, [tex]p \neq2[/tex].
Check p=3
Consider [tex]p=3[/tex]. Then [tex]q=3+70=73[/tex], and [tex]r=73+70=143[/tex]. However, [tex]143=11*13[/tex], so [tex]r[/tex] is not prime. Hence, [tex]p \neq 3[/tex].
So, thus far, we've proven that if there is a prime trio with a common difference of 70, that [tex]p \neq2[/tex] and [tex]p \neq 3[/tex].
Proof of the rest of primes by contradiction
By way of contradiction, assume that there does exist some prime trio [tex]\{ p,q,r \}[/tex] such that [tex]q-p=r-q=70[/tex], and [tex]p \neq2[/tex] and [tex]p \neq 3[/tex].
Then, by the Lemma proven earlier, when [tex]p[/tex] is divided by 6, it must have a remainder of 1 or 5.
Case 1: p/6 has remainder 5
Assume that [tex]p[/tex] has remainder 5 when divided by 6. Then, there exists some natural number [tex]n[/tex], such that [tex]p=6n+5[/tex].
[tex]q=p+70\\q=(6n+5)+70\\q=6n+75\\q=3*(2n+25)[/tex]
Since [tex]n[/tex] was a natural number, and the natural numbers are closed over multiplication and addition (meaning, multiplying and adding more natural numbers results in another natural number), then [tex]q[/tex] is divisible by 3.
Since [tex]p[/tex] is prime, [tex]0 < p[/tex], which implies [tex]0+70 < p+70[/tex]
Since [tex]q=p+70[/tex], [tex]70 < q[/tex], so [tex]q[/tex] cannot equal 3. Since [tex]q[/tex] is divisible by 3, but [tex]q \neq 3[/tex], [tex]q[/tex] is not prime, which is a contradiction to the existence of this prime trio.
Therefore, either [tex]p[/tex] cannot have a remainder of 5 when divided by 6, or the prime trio does not exist.
Case 2: p/6 has remainder 1
Assume that [tex]p[/tex] has remainder 1 when divided by 6. Then, there exists some natural number [tex]n[/tex], such that [tex]p=6n+1[/tex].
[tex]q=p+70\\q=(6n+1)+70\\r=q+70\\r=((6n+1)+70)+70\\r=6n+141\\r=3*(2n+47)[/tex]
Since [tex]n[/tex] was a natural number, and the natural numbers are closed over multiplication and addition (meaning, multiplying and adding more natural numbers results in another natural number), [tex]r[/tex] is divisible by 3.
Since [tex]q[/tex] is prime, [tex]0 < q[/tex], implying [tex]0 +70 < q+70[/tex].
Since [tex]r=q+70[/tex], then [tex]70 < r[/tex]. Therefore, [tex]r\neq 3[/tex].
Since [tex]r[/tex] is divisible by 3, but [tex]r\neq 3[/tex], [tex]r[/tex] is not prime, which is a contradiction to the existence of this prime trio (with a common difference of 70). Therefore, either [tex]p[/tex] cannot have a remainder of 1 when divided by 6, or the prime trio does not exist.
Since [tex]p[/tex] was prime, by our lemma [tex]p[/tex] must have either a remainder of 1 or 5. Since both remainder possibilities resulted in a contradiction, our contradiction assumption is false, so there cannot exist a prime trio such that their common difference is 70.
Each exterior angle of a regular decagon has a measure of (3x + 6)°. What is the value of x?
Ox=8
Ox= 10
Ox=13
Ox=18
The elementary school lunch menu repeats every 20 days; the middle school lunch menu repeats
every 15 days. Both schools are serving pizza today. In how may days will both schools serve pizza
again?
Answer:
60 Days.
Step-by-step explanation:
Alright. We're told that the elementary menu repeats every 20 days. All right. And the middle school repeats every 15 days. And today they're both serving pizza. It asks how long until they're both saving it serving pizza again. Alright. So we can kind of just write them out and say well, if it's every 20 days, it repeats. Well then in 20 days from now we're going to get pizza again. And then after that another 20 days, so 40 days and then after that another 60 days. Okay. And we're just adding 20 each time. Uh you know, to see when would we get pizza again at the elementary school? Well, we can do the same thing here with middle school. And when are they going to get pizza again? Well, in 15 days from now they would get pizza again and then another 15 days from that. So that's 30 And then 45 and then 60 and then 75 and so on. And the first time that they're going to repeat is well, this is the common factor that are the common multiple that they have, which is 60. This is really just a complicated way of asking what is the least common multiple of 20 and 15. Alright. And your answer is going to be 60. So it'll be 60 days until they serve the same pizza again.
What is the range of the exponential function shown below?
Answer:
D. y>0.
Step-by-step explanation:
1) the domain of the given function is any real numbers, then
2) if x→+∞, then y→0; if x→-∞, then y→+∞;
3) finally, the required range is y>0; answer D.
Solve for x in the following equation:
42x-5=23x+7
O x=-17
O x=-12
O x = 12
Ox=17
Answer:
x = 17
Step-by-step explanation:
1) Make the bases the same then rewrite.
4 = 2²
2^2(2x - 5) = 2^3x + 7
2) Since the bases are the same, set the exponents equal to each other.
2(2x -5) = 3x + 7
4x - 10 = 3x + 7
4x - 3x = 7 + 10
x = 17
A circle has a radius of 17cm. Find the radian measure of the central angle θ that intercepts an arc of length 5cm.
Answer:
3.14 PIE
Step-by-step explanation:
Use Pie the method we all learned I think it’s 3.14
find sum of 5, 13, 21, ... 181
A banana grows from a tiny flower that is called "a finger". Each row of
bananas is called a "hand" and can contain 14 to 20 "fingers". Each banana
bunch can grow 9 to 12 hands.
1) What is the least amount of bananas that can grow from each bunch?
The smallest number of bananas that can grow in a bunch is 126.
How to calculate the least amount of bananas that can grow in a bunch?To calculate the least amount of bananas that can grow in a bunch, we must identify which are the smallest possibilities, in this case they are 14 fingers per hand and 9 hands per bunch.
According to the above, the least amount of bananas per bunch would be:
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Sadie made a playlist for her road trip to Arizona. She added the same number of songs from each of her 6 favorite bands. In all, Sadie added 72 songs to the playlist.
Answer: 12 each
Step-by-step explanation: 72 divided by 6 = 12
Find the volume of a sphere with a diameter of 3 centimeters
Answer: 14.14
Step-by-step explanation:
Volume=(4/3)*π*r³
Given d=3
as we know that d=2r
3/2=r
1.5=r
Volume=(4/3)*π*(1.5)³
Volume =14.14
Question 10 of 10
Use long division to find the quotient below.
(4x³+ 11x2 +9) + (x+3)
A. 4x²+x+3
OB. 4x²-2x+3
OC. 4x²-x+ 3
OD. 4x²+2x+3
The quotient of the division using the long division rule is 4x^2 - x + 3
Synthetic division of functionsPolynomial functions are functions that has a leading degree of 3 and above. Given the following parameters.
Dividend = 4x³+ 11x2 +9
Divisor = x + 3
Using the synthetic division to determine the quotient
4x^2 - x + 3
x+3 | 4x³+ 11x2 +9
4x³ + 12x²
__________________
-x² + 0x
-x² - 3x
___________________
3x+9
3x+9
___________________
-------
Hence the quotient of the division using the long division rule is 4x^2 - x + 3
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There’s a garden in the neighborhood, and the number of bushes in the garden can be determined by the equation B = 2s, where B is the number of bushes and s is the number of seeds planted. How many seeds were planted if there are 10 bushes?
You work as an assistant for a local musical theatre group, the Main Street Theatre Company. The company is putting on a production of Les Misérables this year, and they need your help planning it.
All the roles have now been cast, and the lead actors playing the following parts are gathered around a large circular table for their first read: Jean Valjean, Inspector Javert, Thénardier, Enjolras, Marius, Fantine, adult Cosette, adult Éponine, and Madame Thénardier. It is your job to seat these 9 actors, the 3 producers, and the director around the table.
Remember that there are 5 lead male roles and 4 lead female roles. If you must seat the male leads together, the female leads together, the 3 producers together, and the director by himself, how many different ways can you seat the 13 people around the circular table?
103,680
479,001,600
6,227,020,800
17,280
Answer:
6227020800
is mi answer
Anu bought 15 stickers for 225 each and Arjun bought 25 stickers for 215 each. Who paid more? Give a reason for your answer.
Answer: Arjun paid more than Anu of about $2000.
Total cost
unit amount × per unit cost
For Anu total cost:
⇒ 15 × $225
⇒ $3375
For Arjun total cost:
⇒ 25 × $215
⇒ $5375
Anu paid $3375 for each whereas Arjun paid $5375 for each.
Conclusion: Arjun paid more.
Can someone help me?
Answer:
m∠R = 112°
m∠S = 112°
m∠T = 68°
Step-by-step explanation:
Quadrilateral QRST is a cyclic quadrilateral.
A cyclic quadrilateral is a quadrilateral drawn inside a circle where every vertex touches the circumference of the circle.
The opposite angles in a cyclic quadrilateral sum to 180°.
⇒ m∠Q + m∠S = 180°
⇒ m∠R + m∠T = 180°
Given:
m∠Q = 68°m∠R = (3x + 40)°m∠T = (5x - 52)°Measure of angle Q
⇒ m∠Q + m∠S = 180°
⇒ 68° + m∠S = 180°
⇒ m∠S = 180° - 68°
⇒ m∠S = 112°
Measure of angles R and T
⇒ m∠R + m∠T = 180°
⇒ (3x + 40)° + (5x - 52)° = 180°
⇒ )8x -12)° = 180°
⇒ 8x° = 192°
⇒ x = 24
Substituting the found value of x into the expressions for angles R and T:
⇒ m∠R = (3(24) + 40)°
⇒ m∠R = 112°
⇒ m∠T = (5x - 52)°
⇒ m∠T = 68°
The table shows the educational attainment of the population of Mars, ages 25 and over, expressed in millions. Find the probability that a randomly selected martian, aged 25 and over has completed four years of high school only or is a male. The probability is?
The probability of picking a martian that has completed 4 years of high school or is a male is 72 / 85.
What is the probability?Probability determines the odds that a random event would happen. The odds of the event happening lie between 0 and 1.
The probability = (total number of people who have finished 4 years of high school / total number of martains) + (total number of males / total number of martains)
58 / 170 + 86/170 = 144 / 170 = 72 / 85
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Justin's box of crayons was full. He gave half the crayons in the box
to Daniel. How much remained in Justin's box?
Use the Histogram below to answer
Considering the given histogram, we have that:
1. Chebyshev's Theorem.
2. At least 89%.
3. At least 75%.
What does Chebyshev’s Theorem state?When we have no information about the population distribution, Chebyshev's Theorem is used. It states that:
At least 75% of the measures are within 2 standard deviations of the mean.At least 89% of the measures are within 3 standard deviations of the mean.An in general terms, the percentage of measures within k standard deviations of the mean is given by [tex]100(1 - \frac{1}{k^{2}})[/tex].
We can see from the histogram that the distribution is not normal, hence Chebyshev's Theorem can be used.
-15 and 105 are within 3 standard deviations of the mean, hence the percentage is of at least 89%.
As stated in the bullet points, at least 75% of the measures are within 2 standard deviations of the mean.
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Choose the most convenient method to graph the line y=−13x+5
Select the correct answer below:
Recognize the equation as that of a vertical line passing through the x-axis at 5
Recognize the equation as that of a horizontal line passing through the y-axis at 5
Identify the slope and y-intercept and then graph
Find the x- and y-intercepts and then graph
The most convenient method to graph the line y=−13x+5 is C. Identify the slope and y-intercept and then graph.
How to illustrate the information?From the information given, the equation has been given. It's important to identify the slope. This is the gradient of the line.
Also, the y intercept should be identified. Therefore, the most convenient method to graph the line y=−13x+5 is to identify the slope and y-intercept and then graph.
In conclusion, the correct option is C.
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When 5times a number is increased by 10, the answer is the same as when 100 is decreased by the number. What is the number?
Answer:
x = 15
Step-by-step explanation:
[tex]5x + 10 = 100 - x[/tex]
[tex]6x = 90[/tex]
[tex]x = 15[/tex]
12
intent attribution
QUESTION 5 1 POINT
.
Tickets for a basketball game cost $3 for students and $5 for adults. The number of students was 11 less than 8 times
number of adults. The total amount of money from ticket sales was $895. How many of each ticket were sold?
Provide your answer below:
adult tickets,
FEEDI
student tickets
Answer:
32 adult245 studentStep-by-step explanation:
The given relations between numbers and prices can be used to write a system of equations for the number of each kind of ticket. One equation will describe ticket numbers; the other will describe ticket revenue.
SetupLet x represent the number of adult tickets, and y the number of student tickets. The given relations can be written as the general-form equations ...
8x -y -11 = 0 . . . . . . . . . ticket numbers
5x +3y -895 = 0 . . . . . ticket revenue
SolutionThe setup using general form equations lends itself to solution using the "cross multiplication method." Using this method, we define three numbers used in the solution:
Δ = 8(3) -5(-1) = 24 +5 = 29
Δx = -1(-895) -3(-11) = 895 +33 = 928
Δy = -11(5) -(-895)(8) = -55 +7160 = 7105
And the solutions are ...
x = Δx/Δ = 928/29 = 32
y = Δy/Δ = 7105/29 = 245
32 adult tickets and 245 student tickets were sold.
__
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1. What is the Importance of Finance in a Business?
2. What is the impact of Business in our economy?
Answer:
1) Finance is very important in a business as to meet the expenses from manufacturing a product to solding it capital is required. without finance one cannot run a business . even a small scale business requires some finance.
2) Business has a great impact in our economy. it increases the overall economy and contributes somewhere between 70 to 72% to the GDP.
Then what is the value of x?
Answer:
d.) x = -40
Step-by-step explanation:
[tex]\sf Given \ equation: \ log_{5} (-3x+5) = 3[/tex]
Solving steps:
[tex]\rightarrow \sf log_{5} (-3x+5) = 3[/tex]
If logₐ(b) = c then b = aᶜ
[tex]\rightarrow \sf -3x+5 = 5^3[/tex]
[tex]\rightarrow \sf -3x+5 = 125[/tex]
[tex]\rightarrow \sf -3x = 125-5[/tex]
[tex]\rightarrow \sf -3x = 120[/tex]
[tex]\rightarrow \sf x = 120/-3[/tex]
[tex]\rightarrow \sf x = -40[/tex]
Answer:
x= -40
Hope this helps!!
Select three ratios that are equivalent to 11:111:111, colon, 1.
Choose 3 answers:
The three ratios that are equivalent to the given ratio are options B, C and E.
How to determine the equivalent ratios?In order to determine the ratios that are equivalent to the given ratio, we would evaluate by dividing each of them by their respective denominator value:
(11:111:111)/1 = 11:111:111.
For choice A, we have:
1:111:111/11 = 1/11:111:111 (not equivalent).
For choice B, we have:
22:222:222/2 = 11:111:111. (equivalent).
For choice C, we have:
55:555:555/5 = 11:111:111. (equivalent).
For choice D, we have:
9:999:999/99 = 1/11:111:111 (not equivalent).
For choice E, we have:
110:1110:1110/10 = 11:111:111. (equivalent).
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Complete Question:
Select three ratios that are equivalent to 11:111:111, colon, 1. Choose 3 answers: Choose 3 answers: (Choice A) A 1:111:111, colon, 11 (Choice B) B 22:222:222, colon, 2 (Choice C) C 55:555:555, colon, 5 (Choice D) D 9:999:999, colon, 99 (Choice E) E 110:1110:1110, colon, 10.
What is the measure of n?
28
n = [?]
n
7
3
Answer: 14
Step-by-step explanation:
By the geometric mean theorem,
[tex]\frac{n}{28}=\frac{7}{n}\\\\n^{2}=196\\\\n=\boxed{14}[/tex]
The degree of polynomial, x^4-x^2+2 is
Answer:
4
Step-by-step explanation:
You are flying a kite and have let out 30 feet of string but it got caught in an 8ft tree. What is the angle of elevation to the location of the kite? Show your work finding the angle of elevation and state the measure of the angle. You are flying a kite and have let out 30 feet of string but it got caught in an 8ft tree . What is the angle of elevation to the location of the kite ? Show your work finding the angle of elevation and state the measure of the angle .
Answer:
15.47 degrees
Step-by-step explanation:
I need help with number 15 and 16
Answer:
please see explanations
Step-by-step explanation:
For the first question,
Sum of angles in a triangle equal180°3x + 4x + 2x = 180
9x = 180°
Dividing bothsides by 9
9x/9 = 180/9
x = 20
The measure of the interior angles are :
3x = 3(20) = 60
4x = 4( 20) = 80
2x = 2(20) = 40
For the second question,
Let's find x first.
Angles on a straight line equal 180°.x + 92 = 180
x = 180 - 92 = 88°
To find A note that:
Exterior angle equal sum of opposite interior angles.A = 88° + 43°
A = 131°
In this activity, you will use multiple representations of relationships to identify key features and solve problems.
Ryan conducted a 6-day study observing the effects of an organic plant food on the growth of his sprouting bean plant. He tracked these two pieces of information:
the amount of plant food remaining in the container after each day’s feeding
the height of the plant over time
Part A
Ryan found that the amount of plant food remaining decreased an equal amount each day, and he used the entire 72 milliliters by the end of his study.
Part B
Ryan also found that he could model the height of his plant with the equation h = 0.5d + 4, where d is the number of days and h is the height in centimeters.
Part C (This is the question that I need help on....PLEASE?)
Explain the similarities and differences between the two relationships in this situation.
The domain will be 0<x<6 and the range will be 0<y<72.
How to illustrate the information?The information given implies that this would be a continuous relationship.
The process of elimination can be used to rule out the sets of numbers that are used in discrete relationships.
X is the amount of days that he was studying the plants.
The inequality is written like this for x: 0<x<6.
For the range, y is the plant food that was used which is 72 milliliters.
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Solve x2 – 8x + 15 < 0. Select the critical points for the inequality shown. –15 –5 –3 3 5
The critical points are within rthe range 3<x<5
Quadratic equationsThese are equations that has a leading degree of 2. Given the expression below.
x²-8x+15<0
Factorize
x²-3x-5x+15<0
Factor out the GCF
x(x-3)-5(x-3) <0
Group
(x-3)(x-5)<0
Find the solution
x < 3 and x <5
Hence the critical points are within rhe range 3<x<5
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Solve the proportion 2:x x 4:34
[tex]\Large\maltese\underline{\textsf{Our problem:}}[/tex]
Solve the proportion. [tex]\bf{\dfrac{2}{x}=\dfrac{4}{34}[/tex].
[tex]\Large\maltese\underline{\textsf{This problem has been solved!}}[/tex]
We multiply 2 by 34, and x by 4 (this is known as cross multiplication)
[tex]\bf{2*34=4*x}[/tex] | simplify
[tex]\bf{68=4x}[/tex] | solve in terms of x
[tex]\bf{17=x}[/tex]
[tex]\rule{300}{1.7}[/tex]
[tex]\bf{Result:}[/tex]
[tex]\bf{17=x}[/tex]