3. (a) Find the greatest common divisor of 34 and 89 using the Euclidean algo rithm.
(b) Express gcd (34, 89) as a linear combination of 34 and 89.
(c) Find an inverse of 34 modulo 89.
(d) Solve the linear congruence 34x = 53(mod 89).
Answers
a. We have GCD(34, 89) = 1; using the Euclidean algorithm,
89 = 2•34 + 21
34 = 1•21 + 13
21 = 1•13 + 8
13 = 1•8 + 5
8 = 1•5 + 3
5 = 1•3 + 2
3 = 1•2 + 1
b. Working backwards,
1 = 3 - 2
1 = 3 - (5 - 3) = 2•3 - 5
1 = 2•(8 - 5) - 5 = 2•8 - 3•5
1 = 2•8 - 3•(13 - 8) = 5•8 - 3•13
1 = 5•(21 - 13) - 3•13 = 5•21 - 8•13
1 = 5•21 - 8•(34 - 21) = 13•21 - 8•34
1 = 13•(89 - 2•34) - 8•34 = 13•89 - 34•34
c. Using the linear combination find in part b,
1 ≡ 13•89 - 34•34 (mod 89)
1 ≡ (-34)•34 (mod 89)
and
-34 ≡ -34 + 89 ≡ 55 (mod 89)
So, the inverse of 34 modulo 89 is 55.
d. Multiply both sides of the congruence by the inverse of 34:
55•34x ≡ 55•53 (mod 89)
x ≡ 2915 ≡ 32•89 + 67 ≡ 67 (mod 89)
Related Questions
My number has 3 digits. The sum of the digits is 15. The digits are all different odd numbers. If divided by 0.25 the answer is a whole number. The number is greater than 3 and less than 4.
Answers
Answer:
3.75
Step-by-step explanation:
trust me
thanks in advance ill mark brainliest
Answers
Answer:
100º
Step-by-step explanation:
Cooresponding angles are congruent
11x + 1 = 10x + 10
Subtract 10x from both sides
x + 1 = 10
subtract 1 from both sides
x = 9
θ = 11x + 1
θ = 11(9) + 1
θ = 99 + 1
θ = 100º
Please help me out with this Math sheet. Thanks
Answers
Answer:
1. acute
2. obtuse
3. right
4. acute
5. straight
6. obtuse
7. acute
8. obtuse
9. straight
10. ∠ EAD , ∠ CED , ∠ BCE , ∠ BEA , ∠EBC
11. ∠CDA , BCD
12. ∠AED ,∠BEC
13. ∠ AEC, ∠ BED
14. ∠ EAU , ∠EUI , ∠IUO , ∠OAE
15. ∠ EUO , ∠IUA
16. ∠AUO , ∠EIO
Help show work please.
Answers
Answer:
X = 7/3y - 10/3
Step-by-step explanation:
Add - 7y to both sides = -3x + 7y = 10 - 7y
= -3x = 10 - 7y
Divide both sides by -3
-3x/-3 = 10/-3 - 7y/-3
x = 7/3y - 10/3
Max and Toni’s flower bed is rectangular in shape. The length of the garden is 4 feet more than its width and the area of the flower bed is 96 square feet. If x represents the width of the flower bed, find the length and width of the old flower bed. Solve the quadratic equation using the quadratic formula. (20 points)
Answers
Answer: Width = 8ft, Length = 12 ft
Step-by-step explanation:
irst, let'Fs express the width of the garden as w (the unknown that we are trying to solve), since the length is 4 feet longer than it's width, we can express this as 4+w. Since the flowerbed is rectangular, the formula for the area is A=lw where l is it's length and w is it's width, replacing l with 4+w, then replace A for 96 sq.ft and we can begin solving for the unknown w. Multiplying 4+w and w, using the distributive property where a(b+c)=ab+ac, we will wind up with w^2 + 4w = 96, subtracting 96 and moving it to the other side, we end up with w^2 + 4w -96 = 0, now let's use the quadratic formula for the x intercepts where [tex]x = \frac{-b +/- \sqrt{b^2-4ac}}{2a}[/tex] , remember the quadratic equation has the form [tex]ax^2+bx+c[/tex], therefore our a value is 1, our b value is 4, and our c value is -96 (remember to put that negative!), we will get 2 answers, 8 and -12, a negative width is unreasonable and so 8 is the correct width, and since the length is 4 extra feet, this makes the length 12 feet for a total area of 96 feet. hope this helps! Its helpful to draw and visualize the problem!
A large drink dispenser used at a school field day holds 640 ounces of lemonade. How many glasses of lemonade can be poured from the dispenser if each glass holds 8 ounces?
Answers
Answer:
80 glasses
Step-by-step explanation:
640/8 = 80
Answer: 80 glasses
There are 80 glasses of lemonade can be poured from the dispenser if each glass holds 8 ounces.
What is Division method?
Division method is used to distributing a group of things into equal parts. Division is just opposite of multiplications.
For example, dividing 20 by 2 means splitting 20 into 2 equal groups of 10.
Given that;
A large drink dispenser used at a school field day holds 640 ounces of lemonade.
Now,
Find the number of glasses of lemonade can be poured from the dispenser if each glass holds 8 ounces as;
= 640 ÷ 8
= 80
Thus, There are 80 glasses of lemonade can be poured from the dispenser if each glass holds 8 ounces.
Learn more about the divide visit:
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Which expression is equivalent to 4w (-6w)
0 -2w3
O-202
0-24w3
0-24w2
Answers
Step-by-step explanation:
4w(-6w)
= (4*6)(w.-w)
= 24 * -w²
= -24w²
Write a related function for –6x + 8x – 5 = 3.
Answers
Answer:
Answer:
x=4
Step-by-step explanation:6x+8x=2x
5+3=8 (when you move a number to the other side of an equals sign, it changes its value)
2x=8
2x/2 and 8/2
x=4
The related function for –6x + 8x – 5 = 3 is;
x = 4
Algebraic ExpressionsWe are given the equation;
–6x + 8x – 5 = 3
Let us simplify the numbers that have the variable x first. Thus;
–6x + 8x – 5 = 3 gives;
8x - 6x - 5 = 3
>> 2x - 5 = 3
Use addition property of equality to add 5 to both sides;
2x - 5 + 5 = 3 + 5
>> 2x = 8
Use division property of equality to divide both sides by 2 to get;
x = 8/2
Thus;
x = 4
Read more about algebra expressions at; https://brainly.com/question/4344214
Work out the missing numbers
-6x_= -30
-6x_= 0
-6x _=18
_x-6= -54
Answers
Answer:
1. 5
2. 0
3. -3
4. -38
Step-by-step explanation:
from 1 to 4 i answered and solved hope its helpful
how do you simplify 18x⁶/27x⁴
Answers
Answer:
2x^2/3
Step-by-step explanation:
In pic
(Credits: Symbolab)
(Hope this helps can I pls have brainlist (crown)☺️)
What is the area of the following circle?
Either enter the exact answer in terms of π or use 3.14 for π and enter your answer as a decimal.
Answers
Answer:
25π or 78.5
Step-by-step explanation:
Area of circle = π[tex]r^{2}[/tex]
r = 5
A = π * 5^2
= 25π
= 78.5
Jorge wants to find the height of the house, help him find it.
Answers
Answer:
:)
Step-by-step explanation:
:)
What is the area of the figure at the right explain how you found your answer
Answers
Answer:
You have a photo to show the work?
Step-by-step explanation:
Explain how you can multiply to find equivalent fractions
Answers
Answer:
One example for an equivalent fraction would be 2/5 = 4/10
Step-by-step explanation:
It is equivalent because in both the denominator and the numerator you multiply by 2 which means that if you would divide 4/10 by 2 top and bottom, you would get 2/5. If you were to but these fractions on a number line; one on top of the other; they would equally have one under the other.
*** I hope this helps (brainly if possible)
I need the solution urgently plz help me with the question I will mark you brainleist
Answers
Answer:
This is the answer thanks me later
positive and negative relationships are both examples of linear relationships
A. True
B false
Answers
Answer: A. True.
Step-by-step explanation:
A relationship is linear when the points on a scatterplot follow a somewhat straight line pattern. This is the relationship that we will examine. Linear relationships can be either positive or negative. Positive relationships have points that incline upwards to the right. As x values increase, y values increase.
Linkin Corporation is considering purchasing a new delivery truck. The truck has many advantages over the company’s current truck (not the least of which is that it runs). The new truck would cost $55,440. Because of the increased capacity, reduced maintenance costs, and increased fuel economy, the new truck is expected to generate cost savings of $8,400. At the end of 8 years, the company will sell the truck for an estimated $27,700. Traditionally the company has used a rule of thumb that a proposal should not be accepted unless it has a payback period that is less than 50% of the asset’s estimated useful life. Larry Newton, a new manager, has suggested that the company should not rely solely on the payback approach, but should also employ the net present value method when evaluating new projects. The company’s cost of capital is 8%.
Answers
The payback period of the investment is 6.6 years.
The net present value is $7,797.22.
The payback period calculates the amount of time it takes to recover the amount invested from the cumulative cashflows.
Payback period = Amount invested / cost saving
$55,440 / $8,400 = 6.6 years
Net present value is the present value of after-tax cash flows from an investment less the amount invested.
NPV can be determined using a financial calculator
Cash flow in year 0 = $-55440
Cash flow in year 1 to 7 = $8,400
Cash flow in year 8 = $8400 + $27,700
I = 8%
NPV = $7,797.22
To learn more about net present value, please check: https://brainly.com/question/13739943?referrer=searchResult
Given the triangle below, find the value of x
Answers
Answer:
I think it is b I’m not quite sure though. Sorry if wrong…
Step-by-step explanation:
A line has a slope of -4. The line passes through the point (-2,1). What is the point- slope form of the line?
y - 1 = -4 (x + 2)
y - 4 = -2 (x - 1)
y - 1 = 4 (x - 2)
y - 2 = -4 (x - 1)
Answers
Answer:
the answer is A
I followed the point slope formula and added in the x and y values
you will be given ultimate cookie and 20 pts
Answers
Answer: -4, -3, 0, 2
Step-by-step explanation:
Answer:
-4,-3,0,2
Step-by-step explanation:
Decimals that are higher is the less than the smallest
0.5 ÷ 10.2 If you solve this, can you show the steps of this equation?
Answers
Answer:
0.5/ 10.2
Well its basically
Well first you gotta make sure the 10.2 is a whole number in order to divide
so we can multiply by 10
that turns it into 102
Now what we do do 10.2 we do to 0.5 to make it equal which makes0.5*10=5
Now its easier so its 5/102
Now we just divide normally
*assuming its rounded to the nearest hundredths
0.05
Find the equation of the line given the following
Answers
Answer:
1. y = -10x + 6
2. y = -x - 6
3. y = 13x + 110
4. y = -5x + 47
5. y = 7/4 x -7
Step-by-step explanation:
2. m = - 1 and Y-intercept: -20 - (- 1)(14) = -6
3. m = 13 and Y-intercept: -7 - (13)(- 9) = 110
4. (10,-3) and (7,12)
Slope = (12 - - 3)(7- 10) = 15/-3 = -5
y- intercept = 12 - (-5)(7) = 47
5. (4,0) and (0, -7)
Slope: (-7-0)/(0-4) = -7/-4 = 7/4
y-intercept: -7 - 7/4)(0) = -7
Select the correct answer.
Solve the following inequality.
-8 < 3x + 1 < 7
A.
B.
C.
D.
Answers
Answer:
-3 < x < 2
Step-by-step explanation:
-8 < 3x + 1 < 7
-8 < 3x + 1 ....... eqn 1
3x + 1 < 7 ...........eqn 2
-8 < 3x + 1
-8 - 1 < 3x
-9 < 3x
-9/3 < x
-3 < x
3x + 1 < 7 ...........eqn 2
3x < 7 - 1
3x < 6
x < 6/3
x < 2
= -3 < x < 2
PLEASE HELP WITH THE FOLLOWING
Answers
Answer:
2a:
[tex]a^6+\frac{64}{125} =(a^2)^3+(\sqrt[3]{\frac{64}{125} })^3[/tex]
Now, we can apply the sum of cubes formula that goes:
[tex]x^3+y^3=(x+y)(x^2-xy+y^2)[/tex]
Therefore,
[tex](a^2)^3+(\sqrt[3]{\frac{64}{125} })^3=(a^2+\frac{4}{5})(a^4-\frac{4a^2}{5}+(\frac{4}{5} )^2)[/tex]
[tex]=(a^2+\frac{4}{5})(a^4-\frac{4a^2}{5}+\frac{16}{25} )[/tex]
---------------------------------------------
2b:
[tex]16x^8-y^4=(4x^4)^2-(y^2)^2[/tex]
Now, we can apply the difference of two squares formula that goes:
[tex]x^2-y^2=(x+y)(x-y)[/tex]
Therefore,
[tex](4x^4)^2-(y^2)^2=(4x^4+y^2)(4x^4-y^2)[/tex]
You can apply the difference of two squares formula again:
[tex](4x^4+y^2)(4x^4-y^2)=(4x^4+y^2)((2x^2)^2-y^2)[/tex]
[tex](4x^4+y^2)(4x^4-y^2)=(4x^4+y^2)(2x^2+y)(2x^2-y)[/tex]
A camera is on sale for $130. After a 7.875% sales tax is added, what is the total cost of the camera?
Answers
B. DIRECTION: Read and analyze each problem. Give the best answer you can give in order to solve the problem
1. what is asked in the problem?____________________
2. What operation are you going to use to slove the problem?________
3. Show your solution___________________
Answers
Answer:
1: How much longer did he run than walk.
2: Subtraction
3: 1/8 hour
Step-by-step explanation:
1: The last sentence says what is being asked.
2: We want to see how much longer something is so it is subtraction.
3: 3/4 = 6/8 7/8 - 6/8 = 1/8
Type the correct answer in each box. Use numerals instead of words. If necessary, use / for the fraction bar(s).
Answers
Answer:
f^-1(x) = [ln (x + 4)]/2
Step-by-step explanation:
f(x) = e^(2x) - 4
y = e^(2x) - 4
x = e^(2y) - 4
x + 4 = e^(2y)
ln (x + 4) = ln (e^(2y))
ln (x + 4) = 2y
y = [ln (x + 4)]/2
f^-1(x) = [ln (x + 4)]/2
The larger of 2 numbers is 2 more than five times the smaller number. if the sum of the two numbers is 80, find both numbers
Answers
Answer:
The larger number is 67 and the smaller number is 13
Step-by-step explanation:
y=5x+2
x+y=80
-5x+y=2
6x=78
x=13
y=65+2
y=67
A line has a slope of 5 and passes through the point (1,5). What is its equation in
slope-intercept form?
Answers
Answer:
[tex]y=5x[/tex]
Step-by-step explanation:
Insert the information into the point-slope formula to find slope-intercept form.
[tex]y-y_1=m(x-x_1)\\\\y-5=5(x-1)\\\\y-5=5x-5\\\\y=5x[/tex]
Write an equation in slope-intercept form of the line shown.