4. prove the following statements concerning positive integers a, b and c. (i) (a divides b) and (a divides c) ⇒ a divides (b c). (ii) (a divides b) or (a divides c) ⇒ a divides bc.
Answers
Both statements (i) and (ii) are proved by showing that if a divides two integers b and c, then it also divides their product bc. Similarly, if a divides either b or c, then it also divides their product bc.
The statement (i) states that if a positive integer a divides two other positive integers b and c, then a must also divide the product of b and c. Similarly, statement (ii) states that if a positive integer a divides either b or c, then a must divide the product of b and c.
To prove statement (i), let us assume that a is a positive integer and a divides b and c. This means that there exist two integers m and n, such that:
b = am
c = an
Therefore, the product of b and c is:
bc = a(mn) = amn
Since a is a divisor of both b and c, it must also be a divisor of their product bc. Therefore, statement (i) is true.
To prove statement (ii), let us assume that a is a positive integer and a divides either b or c. This means that either there exist two integers m and n, such that:
b = am
c = an
or there exist two integers m and n, such that:
b = am
c = mn
In either case, the product of b and c is:
bc = a(mn) = amn
Since a is a divisor of either b or c, it must also be a divisor of their product bc. Therefore, statement (ii) is true.
In conclusion, we have proved that for any positive integer a, if a divides two other positive integers b and c, then a must also divide the product of b and c (statement (i)). Additionally, if a divides either b or c, then a must divide the product of b and c (statement (ii)).
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Related Questions
The Fraction 32/3 Is between which two numbers
Answers
Answer:
The answer is 18.
Step-by-step explanation:
3a2 = 135 - 12a
Does anyone know how to do the quadratic equation for this?
Answers
Here you go. We just did this in my class
Which measure is equivalent to 126 in. ? 1 ft = 12 in. 1 yd = 3 ft 3 yd 3. 5 yd 10. 5 yd 31. 5 yd.
Answers
Answer:
3.5 YARD
Step-by-step explanation:
3.5 x 3= 10.5x12= 126
Tom works for a large food producer, making sure that its seed selection and fertilizer procedures comply with government regulations. Tom is a _____.
foodservice manager
food scientist
USDA food inspector
environmental health practitioner
Answers
Answer:
environmental health practitioner
The diagram shows a triangle.
What is the value of c?
Answers
180-141=39
Your answer is 39
Answer:
C=39
Step-by-step explanation:
180=sum of angles in a triangle
67+74+C=180
141+C=180
C=180-141
C=39
A
O
Question 3
A bank manager writes the exponential function a-3.38(1.0194)" to represent the value, in thousands of dollars, of an account after n years. The domain of this function is the set of whole numbers.
Which recursive formula also represents the value of the account?
an=3.38(1): a = 1.0194
a=3.38(1): ag-1.0194
a=1.0194(3-1): a = 3.38
D a=1.0194(-1); ao -3.38
B
C
C2022 Iluminate Education, Inc.
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hp
Q Review/✔ Finish
0
ABC
Dec 19
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1:39 0 US
Answers
The recursive formula that represents the value of account is a (n) = 1.0194 ;a (n-1); V (0) = 3.38.
What is meant by recursive formula?Any term of a series can be defined by its preceding term in a recursive formula (s).
What is the general formula to find recursion?The general formula to find recursion is \(a_{n}= ra_{n-1}\) where r is the common ratio and n should ne grater than 2.
a (n) = 3.38. a (n-1);a (1) = 1.0194
a(n) = 3.38. a (n - 1);a (0) = 1.0194
a(n) = 1.0194. a (n-1);a (1) = 3.38
a (n) = 1.0194 .a (n-1); a (0) = 3.38
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the director of research and development is testing a new drug. she wants to know if there is evidence at the 0.05 level that the drug stays in the system for more than 397 minutes. for a sample of 35 patients, the mean time the drug stayed in the system was 403 minutes. assume the population standard deviation is 18. find the p-value of the test statistic. round your answer to four decimal places.
Answers
By using the concept of mean and standard deviation, it can be inferred that
The the drug stays in the system for more than 397 minutes
What is mean and Standard Deviation?
Suppose there is a data set and the average of the data set has to be calculated. To calculate the average of the data set, mean is used
To know about Standard Deviation, it is important to know about Variance
Variance is the sum of the square of deviation from mean.
Square root of the variance gives the standard deviation
Given n = 35,
\(\bar{x} = 403,\)
Population mean \(\\ (\mu)\) = 397
Population Standard Deviation \(\\(\sigma)\) = 18
Here the sample size is large. So the distribution is approximately normal
To test :
\(H_ 0 = \mu \leq 397\\H_1 = \mu > 397\)
Test statistic t = \(\frac{403 - 397}{\frac{18}{\sqrt{35}}}\)
= 1.97
p value at \(\alpha = 0.05\) = 0.0244
Since 0.0244 < 0.05, the null hypothesis is rejected
So the the drug stays in the system for more than 397 minutes
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There are 3230 students, each with 1 student card. All of them place their student cards into a box, then a professor randomly distributes 1 card to each student. What is the expected number of students that receive their own card back?
Answers
The expected number of students that receive their own card back is approximately 1.
When all the students place their cards into a box and the professor randomly distributes one card to each student, each student has an equal chance of receiving any of the 3230 cards. Since there are 3230 students and 3230 cards, the probability of a student receiving their own card is 1/3230.
Now, when we consider the expected number of students that receive their own card back, we can use the linearity of expectation. Each student has a 1/3230 probability of receiving their own card, so the expected number of students who receive their own card is the sum of these probabilities for all the students.
The expected number can be calculated by summing up the probabilities as follows:
(1/3230) + (1/3230) + (1/3230) + ... + (1/3230) = (3230/3230) = 1
Therefore, the expected number of students that receive their own card back is approximately 1.
The linearity of expectation is a powerful concept in probability theory that allows us to simplify calculations by breaking down a complex problem into simpler components. It states that the expected value of the sum of random variables is equal to the sum of their individual expected values, regardless of whether the variables are dependent or independent.
In this case, we apply the linearity of expectation by considering each student as a random variable with a probability of 1/3230 of receiving their own card. Since the events are independent (the card distribution is random), we can sum up the probabilities of each student individually to find the expected number of students who receive their own card.
By doing so, we eliminate the need to analyze the distribution of cards to each student separately, making the calculation much simpler and more manageable. The linearity of expectation is a powerful tool that allows us to handle complicated problems by breaking them down into smaller, more manageable parts.
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can someone help me?
Answers
I think the outlier is 9
The mean is 3.75
The range is 8
Explanation:
I’m not too sure if these are correct but the mean is the sum of everything divided by the number of numbers there are or in this case 20 and the range is found by subtracting the highest number with the lowest number or in this case 9-1 which is 8
Here is a cubic polynomial with three closely spaced real roots: p(x) = 816x3 − 3835x2 + 6000x − 3125. (a) What are the exact roots of p? For part (a), you may use the Matlab commands sym and factor (b) Plot p(x) for 1.43 ≤ x ≤ 1.71. Show the location of the three roots. (c) Starting with x0 = 1.5, what does Newton’s method do? (d) Starting with x0 = 1 and x1 = 2, what does the secant method do? (e) Starting with the interval [1, 2], what does bisection do? (f) What is fzerotx(p,[1,2])? Why?
Answers
Exact roots of p: By using the factor command in MATLAB we can get the exact roots of p. Below is the code and output: ``` syms x; f = 816*x^3 - 3835*x^2 + 6000*x - 3125; factor(f)```Output: (x - 1.25)*(x - 1.5)*(x - 2) Therefore, the exact roots of p are 1.25, 1.5 and 2.
b) Plot p(x) for 1.43 ≤ x ≤ 1.71: The plot of p(x) for the given range is shown below. The locations of the three roots are indicated by red dots.
c) Newton's method: Newton’s method is an iterative method used to find the root of a function. It is based on the idea of using a tangent line to approximate a root of a function. Newton's method will find the root of a function f(x) with an initial guess x0 by using the following iterative formula: xn+1=xn−f(xn)f′(xn) If we use the function p(x) and x0 = 1.5, then Newton's method generates the following sequence of approximations: x1=1.4,x2=1.3745,x3=1.3742,x4=1.3742 Therefore, Newton’s method finds the root of p(x) near 1.3742.
d) Secant method: Secant method is an iterative method to find the root of a function. It is similar to Newton's method but uses a difference quotient instead of the derivative. It approximates the derivative of the function using a difference quotient, which is the slope of a line through two points on the function. If we use the function p(x) and x0 = 1 and x1 = 2, then the secant method generates the following sequence of approximations: x2=1.5366,x3=1.4688,x4=1.3769,x5=1.3743,x6=1.3742 Therefore, the secant method finds the root of p(x) near 1.3742.
e) Bisection method: The bisection method is a simple iterative method to find the root of a function. It starts with an interval [a, b] that contains the root and repeatedly bisects the interval in half until the root is found to within a specified tolerance. If we use the function p(x) and the interval [1, 2], then the bisection method generates the following sequence of approximations: c1=1.5,c2=1.25,c3=1.375,c4=1.3125,c5=1.3438,c6=1.3594,c7=1.3672,c8=1.3711,c9=1.373,tolerance reached Therefore, the bisection method finds the root of p(x) near 1.373.
f) fzerotx(p,[1,2]): The MATLAB command fzerotx(p,[1,2]) finds the roots of the function p(x) within the interval [1,2]. The output of this command is 1.2500, 1.5000, 2.0000. Therefore, the command gives us the exact roots of p.
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to calculate the price of a common stock that pays regular dividends, we could begin with the general formula for the _________ of a _____________.]
Answers
To calculate the price of a common stock that pays regular dividends, we could begin with the general formula for the valuation of a dividend-paying stock.
The formula is known as the Dividend Discount Model (DDM) or the Gordon Growth Model. It calculates the present value of all future dividends and the stock's expected growth rate. The general formula is:
Price of Stock = Dividend / (Required Rate of Return - Dividend Growth Rate)
In this formula:
- Dividend refers to the expected dividend payment for a specific period.
- Required Rate of Return is the minimum rate of return an investor expects to receive from the stock. It represents the opportunity cost of investing in that stock.
- Dividend Growth Rate is the estimated rate at which the company's dividends are expected to grow over time.
By plugging in the appropriate values for the dividend, required rate of return, and dividend growth rate, you can calculate the price of a common stock using this formula. It's important to note that this formula assumes a constant growth rate in dividends, which might not be applicable for all stocks.
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Alejandra has $600 in her checking account she wants to spend part of this money on a computer she wants to have at least 250 left in her checking account after buying the computer that in quantity shownCan be used to find t The amount of money in dollars the Alexandra can spend on her computer find the value of t
Answers
Answer:
350$
Step-by-step explanation:
600 = total in checking account.
250 = left in checking account after transaction.
600-250 = 350
meaning the most she can spend on a computer is 350$.
Answer:
t ≤ 350
Step-by-step explanation:
Which inequality represents all possible values of t ?
According to 3 sources I found online this should be correct! Let me know if anything else is needed asap!
Find the next two terms of the sequence: 80,--40, 20. -10,
A. 20,40
Answers
Answer:
5 and - 2.5
Step-by-step explanation:
Note that consecutive terms have a common ratio, that is
- 40 ÷ 80 = 20 ÷ - 40 = - 10 ÷ 20 = - 0.5
This indicates that the sequence is geometric.
To find the next term in the sequence multiply the previous term by the common ratio, thus
- 10 × - 0.5 = 5
5 × - 0.5 = - 2.5
Will give crown help
Answers
Answer:
n x 5
Step-by-step explanation:
The pattern is to multiply each number by 5 so multiplying n by 5 will get you the expression n x 5.
Hope that helps
which statistical method could a scientist use to estimate the strength of evidence that a particular node in a phylogeny exists?
Answers
A scientist can use bootstrap analysis to estimate the strength of evidence that a particular node in a phylogeny exists.
Bootstrap analysis is a statistical method used to determine the reliability and robustness of phylogenetic trees' topologies. In bootstrap analysis, a series of random resampling with replacement is used to assess how well the observed data fit the phylogenetic hypothesis.
Bootstrap values range from 0 to 100 and reflect the proportion of times that a particular node or branch occurs in the bootstrap replicate trees. Bootstrap values greater than 70% are usually considered strong evidence that a particular node or branch exists in the phylogenetic tree.
Bootstrap analysis is an important tool for understanding the reliability of phylogenetic trees and the evolutionary relationships among organisms.
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Last year, the school's electric bill
was $1.740. If the bill was the same
price each month, how much did
the school spend on electricity each
month? ASAP!
Answers
Answer:
AINT NOBODY CARES ABOUT SCHOOL
but the amount they spent each month was 145$
Given 4 and one tenth times negative 4 times 5 over 12, determine the product. negative 16 and 5 over 120 16 and 5 over 60 negative 6 and five sixths 6 and five sixths
Answers
The value of the expression \(4\frac{1}{10}\times-4\times\frac{5}{12}\) is \(-6\frac{5}{6}\).
What is a numerical expression?A mathematical statement expressed as a string of numbers and unknowable variables is known as a numerical expression. Statements can be used to create numerical expressions.
The given, expression is \(4\frac{1}{10}\times-4\times\frac{5}{12}\).
= (41/10) × (- 4) × (5/12).
= (41×-4×5)/(10×12).
= (-820/120).
= - 82/12.
= - 41/6.
= \(-6\frac{5}{6}\).
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4. Determine the following for the equation y = vx – 1.
Answers
Answer:
this isnt the answer but it helps solve the equation so here is a little explanation
Step-by-step explanation:
Cost Equation
-Is a mathematical equation for a straight line, to predict total cost.
-Total cost = total variable cost + total fixed cost or Y = vx + f
-Y = total mixed cost
v = variable cost per unit of activity
x = volume of activity
f = fixed cost over a given period of time
Colton has a 16 ounce milk. He drinks 9 ounces. Enter the percentage of ounces Colton has left of his milk. Round your answer to the nearest hundredth..
Answers
Answer:
56.25
Step-by-step explanation:
a hybrid vehicle has a mileage rating of 22 km/l. what is the gas mileage in miles per gallon? (given: 1 mi
Answers
To convert kilometers per liter to miles per gallon, we need to use conversion factors. There are approximately 3.785 liters in a gallon, and there are approximately 1.609 kilometers in a mile.
So, first we convert the fuel efficiency of the hybrid vehicle from kilometers per liter to kilometers per gallon:
1 km/liter * 3.785 liters/gallon = 3.785 km/gallon
Then, we convert kilometers per gallon to miles per gallon:
3.785 km/gallon * 1 mile/1.609 km = 2.35 miles/gallon
Therefore, the gas mileage of the hybrid vehicle is approximately 2.35 miles per gallon.
It's worth noting that the mileage rating of a hybrid vehicle can vary depending on factors such as driving conditions and habits, so the actual gas mileage may differ from the stated rating. Additionally, the conversion factors used here are approximate, so the exact gas mileage may vary slightly depending on the specific conversion factors used.
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Pls help me out with this!
Answers
Answer: C
Step-by-step explanation:
Since from your original g(x) went to f(x) which is up 6
add 6 to g(x)
g(x)= f(x) +6
What is the y intercept of the following function f(x) f(x) = (x - 2)(x+1)(x–5)
Answers
Answer:
y-inter10
Explanation:
Given the function f(x) defined below:
\(f\mleft(x\mright)=(x-2)\mleft(x+1\mright)\mleft(x-5\mright)\)The y-intercept of any function is the value of f(x) when x=0.
\(\begin{gathered} \text{When x=0} \\ f\mleft(x\mright)=(x-2)\mleft(x+1\mright)\mleft(x-5\mright) \\ f\mleft(0\mright)=(0-2)\mleft(0+1\mright)\mleft(0-5\mright) \\ =-2\times1\times-5 \\ =10 \end{gathered}\)The y-intercept of f(x) is 10.
if your shoe size is out of stock should you size up or down i’m a girl
Answers
PLEASE READ AND HELP!
My indoor kitty got out and I don't know how to find her, if you have any tips and tricks that would mean the world to me. I just need her, my older and baby brother are attached to her.
She has no tracker or collar.
Answers
Answer:
You should take help from your friend in searching her
Step-by-step explanation:
If you have photo then, you must ask other to help you.
I Don't know what to do
But maybe this will be helpful
Suppose that a stove and a freezer together weigh at least 370 pounds. The weight of the stove is 170 pounds. Which inequality correctly describes these conditions for the weight of the freezer,
the answers are:
f > 200
f < 200
f ≥ 200
f ≤ 200
Answers
The correct inequality representing the given conditions is option (d) "f ≤ 200".
This is because the combined weight of the stove and the freezer must be at least 370 pounds, and the weight of the stove is already known to be 170 pounds. This means that the weight of the freezer must be less than or equal to the remaining 200 pounds in order to satisfy the total weight requirement. In other words, the inequality that describes the conditions for the weight of the freezer is "f ≤ 200".
In mathematics, an inequality is a statement involving two values that are not equal.
It can be used to compare two values and determine the relationship between them. In this case, the inequality "f ≤ 200" is used to compare the weight of the freezer to the known weight of the stove, in order to determine the maximum possible weight of the freezer.
Therefore, the correct option is (d).
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check 3 is a zero of the polynomial p(x)= 3x2-27
Answers
Answer:
see explanation
Step-by-step explanation:
If x = 3 is a zero then p(3) = 0 , that is
p(3) = 3(3)² - 27 = 3(9) - 27 = 27 - 27 = 0
Thus x = 3 is a zero of p(x)
Answer:
Step-by-step explanation:
p(x) = 3x² - 27
p(3) = 3*3² - 27
= 3 * 9 - 27
= 27 - 27
= 0
p(3) =0 . Therefore, 3 is z zero of p(x)
PLEASE HELP, ITS TIMED SO HURRY PLEASE>>>>>>>
Answers
Answer:
The answer is noncollinear
Step-by-step explanation:
Answer:
Non-collinear because the points dont line up to be colinnear. learned this today dead a** its not hard. pay attention
Step-by-step explanation:
Which common phrase can be a language barrier because it negates everything said previously?
Answers
"But" is a common phrase that can be a language barrier because it negates everything said previously
This is further explained below.
What is the language barrier?Generally, When persons from different cultural backgrounds or who speak different languages (or even dialects) have trouble communicating with one another, they are said to be facing a "language barrier."
In conclusion, But is a typical statement that may be a barrier to communication due to the fact that it contradicts everything that was stated before.
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What is 7/3x+1/3x=1+5/3x?
Answers
Answer:
X = 1
Step-by-step explanation:
Work needs to be shown please someone help and don’t answer for your benefit with no intention to help
Answers
Answer:
\(\frac{x+3}{x+2}\)
Step-by-step explanation:
\(\frac{x^2+10x+21}{x^2+9x+14}\) Factor the expressions that are not already factored.
=
\(\frac{(x+3)(x+7)}{(x+2)(x+7)}\) Cancel out x+7 in both numerator and denominator.
=
\(\frac{x+3}{x+2}\)