5. [Chinese Remainder Theorem, 10pt] Use the method of the Chinese Remainder Theorem to solve the following problems. Show your work.
a) [6pt] Find x (between 0 and 3279*1072)
such that
x ≡ 1072 (3279), and x ≡ 77 (2303).
b) [4pt] Find x (between 0 and 5696 * 4803 * 4531)
such that
x ≡ 1072 (3279), x ≡ 77 (2303). and x ≡ 4545 (6731).
Answers
a) We want to solve the system of congruences:
x ≡ 1072 (3279)
x ≡ 77 (2303)
First, we find the solutions to the two congruences separately. For the first congruence, we have:
3279 = 5 * 2303 + 674
So we can write:
x ≡ 1072 (3279) ≡ 1072 (5 * 2303 + 674) ≡ 1072 (674) (mod 2303)
We can use the Euclidean algorithm to find the inverse of 674 modulo 2303:
2303 = 3 * 674 + 281
674 = 2 * 281 + 112
281 = 2 * 112 + 57
112 = 2 * 57 - 2
Working backwards, we have:
1 = 3 - 2 * (674 - 2 * (281 - 2 * 112 + 2)) = 7 * 674 - 6 * 2303
So we can multiply both sides of the congruence by 674 and simplify:
x ≡ 1072 (674) (mod 2303)
x ≡ 722 (mod 2303)
Now, we can use the same method to solve the second congruence:
2303 = 29 * 77 + 42
77 = 1 * 42 + 35
42 = 1 * 35 + 7
35 = 5 * 7 + 0
Working backwards, we have:
1 = -1 * 29 + 2 * 7
= -1 * 29 + 2 * (42 - 1 * 35)
= 2 * 42 - 3 * 35
= 2 * 42 - 3 * (77 - 42)
= -3 * 77 + 5 * 42
= -3 * 77 + 5 * (2303 - 29 * 77)
= -152 * 77 + 5 * 2303
So we can multiply both sides of the congruence by 152 and simplify:
x ≡ 77 (152) (mod 2303)
x ≡ 497 (mod 2303)
Now we have two congruences that we can solve using the Chinese Remainder Theorem. We need to find integers a and b such that:
x ≡ a (3279 * 2303)
x ≡ b (674 * 152)
To find a, we can use the formula:
a = (77 * 3279 * 152 + 1072 * 674 * 2303) mod (3279 * 2303)
To find b, we can use the formula:
b = (1072 * 674 * 152 + 497 * 3279 * 2303) mod (674 * 152)
Evaluating these formulas, we get:
a = 2258536
b = 602064
So the solution to the system of congruences is:
x ≡ 2258536 (mod 3279 * 2303)
x ≡ 602064 (mod 674 * 152)
To find the unique solution x between 0 and 3279 * 1072, we can use the formula:
x = a + (b - a) * (3279 * 2303) * (674 * 152)^(-1) mod (3279 * 1072)
where (674 * 152)^(-1) is the inverse of 674 * 152 modulo 3279 * 1072. We can find this inverse using the Euclidean algorithm:
3279 * 1072 = 3 * 674 * 152 + 536064
674 * 152 = 1 * 536064 + 36320
536064 = 14 * 36320 + 4944
36320 = 7 * 4944 + 272
4944 = 18 * 272 + 240
272 = 1 * 240 + 32
240 = 7 * 32 + 16
32 = 2 * 16 + 0
Working backwards, we have:
1 = 2 - 1 * 1
= 2 - 1 * (32 - 2 * 16)
= -1 * 32 + 3 * 16
= -1 * 32 + 3 * (240 - 7 * 32)
= 22 * 32 - 3 * 240
= 22 * (272 - 240) - 3 * 240
= -25 * 240 + 22 * 272
= -25 * (4944 - 18 * 272) + 22 * 272
= 472 *
Related Questions
a tangent to a circle passes through the center of the circle
True
false
sometimes true
Answers
Answer:
False.
Step-by-step explanation:
A tangent is a line that only touches exactly one point of the circle. (The outer edge) :)
Sandra bought 4 pounds of sweet potatos for $6.12 what is the cost per pound of sweet potatoes
Answers
Answer:
$1.53 per pound
Step-by-step explanation:
if she bought 4 pounds for 6.12 dollars then you can divide 6.12 by 4 to get the answer for one pound of potatoes and 6.12/4 = 1.53
Hope this helps! :)
Part A
Solve 4,500 ÷ 90. Enter your answer in the box.
Part B
Select all the expressions that are equal to 4,500 ÷ 90.
A. 4,500 ÷ 9 tens
B. 450 ÷ 90
C. 450 ÷ 9
D. 4,500 ÷ 9
E. 4,500 ÷ 90 tens
Answers
Answer:
Step-by-step explanation:
Part A
Solve 4,500 ÷ 90. Enter your answer in the box.
Answer : 50
Part B
Select all the expressions that are equal to 4,500 ÷ 90.
A. 4,500 ÷ 9 tens ✔️
B. 450 ÷ 90 ❌
C. 450 ÷ 9 ❌
D. 4,500 ÷ 9 ❌
E. 4,500 ÷ 90 tens ❌
Free Response - calculator may be used. Please help 50 points and brainliest
Answers
Answer:
the answer would most likely be A
Step-by-step explanation:
Answer:
the answer is b
Step-by-step explanation:
Check the statements below that are FALSE.
Answers
Answer:1 3 5 are faulse
Step-by-step explanation:
select all expression to 3.3.3.3.3.3.3.3
Answers
The equivalent expressions are 3^8 and 3561
Hwo to determine the equivalent expressionFrom the question, we have the following parameters that can be used in our computation:
3.3.3.3.3.3.3.3
The expression "3.3.3.3.3.3.3.3" can be simplified as follows:
3^8
The symbol "^" represents exponentiation, so 3^8 means 3 multiplied by itself 8 times.
This expression when calculated, is:
3^8 = 3 * 3 * 3 * 3 * 3 * 3 * 3 * 3 = 6561
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3. (09. 01 LC)
For f(x) = x2 and g(x) = (x - 5), in which direction and by how many units should f(x) be shifted to obtain g(x)? (5 points)
Up 5 units
Down 5 units
Left 5 units
Right 5 units
Answers
To obtain g(x) from f(x), f(x) should be shifted 5 units to the right.
The function f(x) = x^2 represents a parabola that opens upward. The function g(x) = (x - 5) represents a linear function that is a straight line with a slope of 1 and y-intercept of -5. To transform f(x) into g(x), we need to shift the graph of f(x) in a certain direction.
Since g(x) is obtained by subtracting 5 from x, this means that the entire graph of f(x) should be shifted 5 units to the right. By shifting f(x) to the right, we align the points on the graph of f(x) with the points on the graph of g(x) such that the y-values are the same for corresponding x-values.
Therefore, the correct answer is "Right 5 units." Shifting f(x) 5 units to the right aligns it with g(x) and results in the desired transformation.
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A checkbook balance sheet shows an initial balance for the month of $300. During the month, checks were written in the amounts of $25, $82, $213, and $97. Deposits were made into the account in the amounts of $84 and $116. What was the balance at the end of the month?
Answers
Answer:
your answer is 200 if you add 84 and 116 but is you add 25 82 213 and 97 it will be 417
Step-by-step explanation:
The balance at the end of the month is $83.
Checkbook balance = $300
Amount withdrawn = $25 + 82 + $213 + $97 = $417
Amount deposited = $84 + $116 = $200
Therefore, the balance at the month end will be:
= $300 - $417 + 200
= $83
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The goal of building one-to-many relationships between tables is to minimize _____ data. A. misspelled B. rarely used C. numeric D. redundant
Answers
The goal of building one-to-many relationships between tables is to minimize redundant data.
Data redundancy is a regular occurrence in many businesses and occurs when an identical piece of data is stored in two or more different locations. As more businesses switch from using segregated data to a central repository, they discover that their database is full of inconsistent duplicates of the same entry. Understanding how to effectively manage and track data redundancy can help your business avoid long-term inconsistency problems, even though it can be difficult to reconcile — or even benefit from — redundant data entries.
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The goal of building one-to-many relationships between tables is to minimize redundant data.
Redundant data refers to data that is duplicated across multiple tables, which can lead to data inconsistencies and
increase the amount of storage space required.
By creating one-to-many relationships between tables, data is only stored once in the database, and each instance of
the data in other tables refer back to the original source, reducing the need for redundant data storage.
Therefore, The goal of building one-to-many relationships between tables is to minimize redundant data.
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which statements is true of the function f(x)= x2 + 6x3 - 4 + 2x5?
A. the function is positive for all values of x
B. the end behavior of the graph f(x) is similar to that of g(x) = -x
C.the graph has exactly one x-intercept
D. the function is increasing for all real numbers
Answers
The statements is true of the function f(x)= x^2 + 6x^3 - 4 + 2x^5 is C.the graph has exactly one x-intercept
How to show that the function has one interceptIt is shown that the function has one intercept in the x direction by plotting the graph of the polynomial
The graph showed one x intercept is at the coordinate (0.778, 0)
x intercept refers to the value of x when y is zero
The graph is plotted and attached
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How many ways can four guests sit in a row of six chairs?
Answers
Answer:
360
Step-by-step explanation:
There are 6 spaces and 4 people.
_ _ _ _ _ _
The first person to choose a seat can choose any of the 6 seats.
The second person can choose any of the 5 remaining seats.
The third person can choose any of the 4 remaining seats.
The fourth person can choose any of the 3 remaining seats.
Now, we do 6 x 5 x 4 x 3 to figure out the total number of ways they can sit in the seats.
6 x 5 x 4 x 3 = 360 ways
What is the y-intercept of the line whose equation is y=−3x ?
Answers
Answer:
y=mx+b
b is the y intercept
b=0, so 0 is the y intercept
Step-by-step explanation:
a camper lights an oil lantern at noon and lets it burn continuously. once the lantern is lit, the lantern burns oil at a constant rate each hour. at p.m., the amount of oil left in the lantern is ounces. at p.m., the amount of oil left in the lantern is ounces. based on the average rate of oil burning per hour, how much oil, in ounces, was in the lantern at noon?
Answers
There were 16 ounces of oil in the lantern at noon.
Let's start by defining the variables we know. We'll call the amount of oil in the lantern at noon "x," the rate at which the oil burns "r," and the time elapsed from noon to 2 pm "t." We know that the amount of oil in the lantern at 2 pm is 12 ounces, and at 4 pm, it's 8 ounces.
We can use the rate of oil burning to create an equation relating the amount of oil in the lantern to the time elapsed. The equation is:
x - rt = y
where "y" is the amount of oil in the lantern at any given time after noon. We can solve for "x" by plugging in the values we know at 2 pm:
x - 2r = 12
And at 4 pm:
x - 4r = 8
Now we have two equations with two variables. We can solve for "r" by subtracting the second equation from the first:
2r = 4
r = 2
Now we can plug in "r" to one of the equations to solve for "x." Let's use the first equation:
x - 2(2) = 12
x - 4 = 12
x = 16
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you have the numbers 9,12,16,18,23 if the mean is 18 what is the missing number
Answers
The missing number in the set is 12.
To find the missing number when given a set of numbers and their mean, we can use the formula for calculating the mean (also known as the average). The mean is calculated by summing all the numbers in the set and dividing the sum by the total number of values. In this case, we have the numbers 9, 12, 16, 18, and 23, with a mean of 18.
To find the missing number, we can use the equation:
Sum of all numbers = Mean × Total number of values.
In this case, the sum of all numbers is:
9 + 12 + 16 + 18 + 23 = 78.
We know that the mean is 18 and there are 5 numbers in total. Therefore, we can rearrange the equation to find the missing number:
Missing number = Mean × Total number of values - Sum of all numbers.
Missing number = 18 × 5 - 78 = 90 - 78 = 12.
It's important to note that this method assumes that the set of numbers is complete except for the missing value, and that the mean is calculated correctly. If there are any additional factors or assumptions that need to be considered, they should be stated explicitly to ensure an accurate solution.
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if it takes 9 days for 4 workers to build a garage how long would it take 3 workers
Answers
Answer: 6 days and 18 hours
Step-by-step explanation: 9/4=1.25 1.25x3=6.75 and so on
85 out of 200 workers in a building arrive at 8:00 a.m. What percent of workers in the building do NOT arrive at 8:00 a.m.?
Answers
Answer:
57.5 %
Step-by-step explanation:
First find out the number of workers that do not arrive by 8 am
200 -85 = 115
Take this number divided by the total
115/200 = .575
Change to percent form
57.5 %
Lester bought 6.3 pounds of bologna at the deli. Renee bought 2.1 pounds of bologna.how much times more bologna did Lester buy than Renee
Answers
Answer:
4.2
Step-by-step explanation:
6.3 - 2.1 = 4.2
You recently had a cholesterol panel completed and see that the results for your High Density Lipoprotien (HDL) level comes back with z = -1.7 among people of your stature. Your doctor is going to review the results with you but based on what you know about z-scores you can infer: You have an above average HDL level for people of your stature. The score is negative so this will be good news. The average person of your stature is 1.7 deviations below you. Your HDL level is extremely low for a person of your stature. Х Your HDL level is slightly below average. 19 0/1 point Johnny is saving for retirement and wants to maximize his money. He knows the APR will be the same for both options, but he has a choice of $75 a month for 30 years or $150 a month for 15 years. Which should he choose and why? Unable to determine without the exact APR value. х Both choices will result in the same account balance. O He should choose the choice that deposits money for longer to get the best balance. He should choose the choice that deposits the most money each month because to get the best balance. Only a compound interest account will maximize his balance.
Answers
Answer:
Therefore, over a period of 15 years, the account balance will be higher if Johnny deposits $150 a month than if he deposits $75 a month for 30 years.
Step-by-step explanation:
Johnny should choose the option that deposits $150 a month for 15 years, because this will result in the highest account balance. This is due to the effect of compound interest; when a person invests in an account with compound interest, their balance will increase by more each period due to the interest they earn being added to their principal balance.
Therefore, over a period of 15 years, the account balance will be higher if Johnny deposits $150 a month than if he deposits $75 a month for 30 years.
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2s+5≥8 on a number line
Answers
Answer:
s ≥ 3/2
Step-by-step explanation:
2s + 5 ≥ 8
2s ≥ 8 - 5
2s ≥ 3
s ≥ 3/2
Number Line shown below.
Solution:
Note that:
Given 2s + 5 ≥ 8Simplify the inequality, then graph it.
2s + 5 ≥ 8=> 2s ≥ 8 - 5=> 2s ≥ 3=> s ≥ 3/2Now, graph the line. Graph attached~
Just need 106-110 no work just answers please. I would really appreciate it
Answers
107. 72
108. 23
109. 123
110. 271 (I think??)
In an A.P the term is -10 and the 15th term is 11 and the last term is 41. Find the sum of all terms in this progression.
Answers
This is equivalent to the fraction 1085/2
===============================================================
Explanation:
AP stands for "arithmetic progression", which is another name for "arithmetic sequence"
a1 = -10 is the first term
the 15th term happens when n = 15, so
an = a1 + d*(n-1)
a15 = -10 + d(15-1)
a15 = 14d-10
Set this equal to 11 (the stated 15th term) and solve for d
a15 = 11
14d-10 = 11
14d = 11+10
14d = 21
d = 21/14
d = 3/2
d = 1.5 is the common difference
Let's find the nth term
an = a1 + d(n-1)
an = -10 + 1.5(n-1)
an = -10 + 1.5n - 1.5
an = 1.5n - 11.5
-------------------------------
The last term is 41, so we'll replace the 'an' with that and solve for n
an = 1.5n - 11.5
41 = 1.5n - 11.5
41+11.5 = 1.5n
52.5 = 1.5n
1.5n = 52.5
n = (52.5)/(1.5)
n = 35
So the 35th term is 41.
-------------------------------
We're summing n = 35 terms from a1 = -10 to an = 41
S = sum of the first n terms of arithmetic progression
S = (n/2)*(a1 + an)
S = (35/2)*(-10+41)
S = 542.5
The 35 terms add up to 542.5 which is the final answer
As an improper fraction, this converts to 1085/2
I would appreciate help without guessing :)
Answers
Answer:
a) \(\sqrt{56\) → Definitely not undefined because 56 is a positive real number
b) \(-\sqrt{56\) → Definitely not undefined because 56 is a positive real number (and the negative sign is not under the square root)
c) \(\sqrt{-56\) → Definitely undefined because -56 is a negative number
__
We know that there is no real square root of a negative number because nothing multiplied by itself results in a negative number.
__
d) \(\sqrt h\) → Could be undefined because we don't know the value of \(h\); it could be positive or negative
e) \(-\sqrt {h\) → Could be undefined because we don't know the value of \(h\); it could be positive or negative (and the negative sign doesn't affect this because it is outside the square root)
f) \(\sqrt{-h\) (when \(h\) is positive) → Definitely undefined because the value inside of the square root is negative (a negative times a positive is a negative)
Help me!!!!!! Hanger math..
Answers
Answer:x=3
Step-by-step explanation:
Equation:
5x+2=17
5x=15
x=3
Answer
the circles are 3 numbers each for it to be even
NEED ANSWER FAST
(please explain!!!!)
Answers
Answer:
The left answer
Step-by-step explanation:
They are asking if their is a relationship between the x and y value
for the left one, when the value of y is shown to be the x value divided by 3 as
11/3 =11 over 3, 12/3 = 4, 13/3 = 13 over 3
But the right answer does not seem to have any correlation/relationship between its x and y values
4/9x1/9
(Simplify the answer if possible)
Answers
Given the following confidence interval for a population mean, compute the margin of error, EE. 12.73 < µ < 14.33
Answers
The margin of error according to the given confidence interval is 0.4.
To compute the margin of error, EE, we first need to find the midpoint of the confidence interval. The midpoint is calculated by taking the average of the upper and lower bounds:
Midpoint = (12.73 + 14.33) / 2 = 13.53
Next, we need to find the distance from the midpoint to either the upper or lower bound. Since the confidence interval is symmetrical, we can use either bound:
Distance from midpoint to upper bound = 14.33 - 13.53 = 0.8
Finally, we divide the distance by 2 to get the margin of error:
EE = 0.8 / 2 = 0.4
Therefore, the margin of error for this confidence interval is 0.4. This means that we can be 95% confident that the true population mean falls within a range of 13.53 ± 0.4.
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Two small planes start from the same point and fly in opposite directions. The first plane is flying 45 mph slower than the second plane. In 3 h, the planes are 795 mi apart. Find the rate of each plane.
Answers
The rate of each plane is,
First plane = 110 mph
Second plane = 15 mph
Let x = rate of the slower plane (First plane!)
x + 45 = rate of the faster plane (Second plane!)
The planes fly for 3 hours, where Distance = RT
Distance between the planes = SUM of the distances.
RT + RT= 795 miles
3x + 3(x + 45) = 795
3x + 3x + 135 = 795
6x + 135 = 795
6x = 795 - 135
6x = 660
x = 110 mph First plane
And, x + 45 = 110 + 45 = 15 mph Second plane.
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A storage bin has the shape of a cylinder with a conical top. What is the volume of the storage bin if its radius is
r
=
5. 8
ft, the height of the cylindrical portion is
h
=
7. 9
ft, and the overall height is
H
=
16. 7
ft?
Answers
Total volume = (V_cylinder)+(V_cone)
= 827.64 + 305 .45
=1133.09 ft³
To calculate the volume of the storage bin with a cylindrical portion and a conical top, we need to find the volume of each part and add them together.
Given:
Radius (r) = 5.8 ft
Height of cylindrical portion (h) = 7.9 ft
Overall height (H) = 16.7 ft
First, find the height of the conical top by subtracting the height of the cylindrical portion from the overall height:
Height of conical top (h_cone) = H - h = 16.7 - 7.9 = 8.8 ft
Now, calculate the volume of each part:
1. Volume of cylindrical portion (V_cylinder) = π*r²h = π(5.8²)(7.9) ≈ 827.64 ft³
2. Volume of conical top. (V_cone) = (1/3)π*r²h. (V_cone) = (1/3)π(5.8²)(8.8) ≈ 305.45 ft³
Finally, add the volumes of both parts to find the total volume of the storage bin:
Total volume = V_cylinder + V_cone = 827.64 + 305.45 ≈ 1133.09 ft³
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What is the perimeter of this parallelogram? Please help asap!
Answers
18 units
Step-by-step explanation:
a^2 + b^2 = c^2
3^2 + 4^2 = c^2
9 + 16 = c^2
c^2 = 25
Square root of 25 = 5
5 + 5 = 10
(Opposite sides are congruent)
4 + 4 = 8
10 + 8 = 18
Therefore, the perimeter is 18
For the following the equation f(A,B,C,D)=(A+B+D
′
)(A
′
+C)(C+D) 1. Find the expansion for the function 2. Find the minterms SOP with K-map 5.16 Plot the following function on a Karnaugh map (K-map) and find the minimum sum of products for the function f(A,B,C)=∑m(2,3,5,7) 5.17 Plot the following function on a Karnaugh map (K-map) and find the minimum Sum of products for the function F(A,B,C,D)=A
′
B
′
+CD
′
+ABC+A
′
B
′
CD
′
+ABCD
Answers
We can group the minterms to obtain the minimum sum of products (SOP):
F(A,B,C,D) = A'B' + CD' + ABC + A'B'CD' + ABCD
Expansion of the function f(A,B,C,D):
f(A,B,C,D) = (A+B+D')(A'C)(C+D)
= A'A'C + A'CC' + BA'C + BC + D'A'C + D'CC' + BD'(A'C) + BD'(C+D)
= 0 + 0 + BA'C + BC + 0 + 0 + BD'A'C + BD'C + BD'D
= BA'C + BC + BD'A'C + BD'C + BD'D
Minterms SOP for the function with K-map 5.16:
The truth table for the given function is:
A B C F
0 0 0 0
0 0 1 1
0 1 0 1
0 1 1 0
1 0 0 1
1 0 1 0
1 1 0 0
1 1 1 1
The K-map for this function is:
BC
A 00 01 11 10
0 | 0 1 0 1
1 | 1 0 0 1
We can group the minterms to obtain the minimum sum of products (SOP):
F(A,B,C) = A'B' + A'C + BC
Karnaugh map and minimum sum of products for the function F(A,B,C,D)=A'B'+CD'+ABC+A'B'CD'+ABCD:
The truth table for the given function is:
A B C D F
0 0 0 0 0
0 0 0 1 1
0 0 1 0 0
0 0 1 1 0
0 1 0 0 1
0 1 0 1 0
0 1 1 0 1
0 1 1 1 1
1 0 0 0 0
1 0 0 1 0
1 0 1 0 0
1 0 1 1 0
1 1 0 0 1
1 1 0 1 1
1 1 1 0 1
1 1 1 1 1
The K-map for this function is:
CD
AB 00 01 11 10
00 | 0 1 0 0
01 | 1 0 0 1
11 | 1 1 1 1
10 | 0 0 1 0
We can group the minterms to obtain the minimum sum of products (SOP):
F(A,B,C,D) = A'B' + CD' + ABC + A'B'CD' + ABCD
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