Prove that the value of the expression: (36^5−6^9)(38^9−38^8) is divisible by 30 and 37.
_x30x37
Don't answer if you don't know
Answers
To prove that the expression (36^5−6^9)(38^9−38^8) is divisible by 30, we need to show that it is divisible by both 2 and 3.
First, we can factor out a 6^9 from the first term:
(36^5−6^9)(38^9−38^8) = 6^9(6^10-36^5)(38^9-38^8)
Notice that 6^10 can be written as (2*3)^10, which is clearly divisible by both 2 and 3. Also, 36 is divisible by 3, so 36^5 is divisible by 3^5. Thus, we can write:
6^9(6^10-36^5) = 6^9(2^10*3^10 - 3^5*2^10) = 6^9*2^10*(3^10 - 3^5)
Since 2^10 is divisible by 2, and 3^10 - 3^5 is clearly divisible by 3, the whole expression is divisible by both 2 and 3, and therefore divisible by 30.
To prove that the expression is divisible by 37, we can use Fermat's Little Theorem. Fermat's Little Theorem states that if p is a prime number and a is any positive integer not divisible by p, then a^(p-1) is congruent to 1 modulo p, which can be written as a^(p-1) ≡ 1 (mod p).
In this case, p = 37, and 36 is not divisible by 37. Therefore, by Fermat's Little Theorem:
36^(37-1) ≡ 1 (mod 37)
Simplifying the exponent gives:
36^36 ≡ 1 (mod 37)
Similarly, 38 is not divisible by 37, so:
38^(37-1) ≡ 1 (mod 37)
Simplifying the exponent gives:
38^36 ≡ 1 (mod 37)
Now we can use these congruences to simplify our expression:
(36^5−6^9)(38^9−38^8) ≡ (-6^9)(-1) ≡ 6^9 (mod 37)
We know that 6^9 is divisible by 3, so we can write:
6^9 = 2^9*3^9
Since 2 and 37 are relatively prime, we can use Euler's Totient Theorem to simplify 2^9 (mod 37):
2^φ(37) ≡ 2^36 ≡ 1 (mod 37)
Therefore:
2^9 ≡ 2^9*1 ≡ 2^9*2^36 ≡ 2^(9+36) ≡ 2^45 (mod 37)
Now we can simplify our expression further:
6^9 ≡ 2^45*3^9 ≡ (2^5)^9*3^9 ≡ 32^9*3^9 (mod 37)
Notice that 32 is congruent to -5 modulo 37, since 32+5 = 37. Therefore:
32^9 ≡ (-5)^9 ≡ -5^9 ≡ -1953125 ≡ 2 (mod 37)
So:
6^9 ≡ 2*3^9 ≡ 2*19683 ≡ 39366 ≡ 0 (mod 37)
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Related Questions
The area of a rectangular room is 750 square feet. The width of the room is 5 feet less than the length of the room. Which equations can be used to solve for y, the length of the room? Select three options. y(y + 5) = 750 y2 – 5y = 750 750 – y(y – 5) = 0 y(y – 5) + 750 = 0 (y + 25)(y – 30) = 0
Answers
y2 – 5y = 750
750 – y(y – 5) = 0
(y + 25)(y – 30) = 0
What is the slope of the line that contains these points?
Answers
The slope can be found with two sets of points
The points can be any points, but in this case I choose the point (-1,10) and (1,2)
Slope = \(\frac{y2-y1}{x2-x1} =\frac{10-2}{-1-1} =\frac{8}{-2} =-4\)
The slope of the line that contains these points is -4.
Hope that helps!
Kuong Inc. sold a commercial office building used in the corporate business for $1.5 million. Kuong purchased the building in 2000 for a cost of $1.4 million and had deducted $538,000 MACRS depreciation through date of sale. Kuong should characterize the $638,000 gain recognized on sale as:
Answers
Kuong Inc. should characterize the $638,000 gain recognized on the sale of the commercial office building as a combination of depreciation recapture and capital-gain after deducting MACRS-depreciation.
Kuong should characterize the $638,000 gain recognized on sale as the $538,000 of MACRS depreciation deducted will be treated as depreciation recapture, which is generally taxed at a higher rate than capital gains. The remaining $100,000 gain (sale price of $1.5 million minus the adjusted basis of $1.4 million) is considered a capital gain, which may be subject to a lower tax rate depending on the holding period and other factors.
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Zoe is decorating the outside of a box in the shape of a triangular prism. The figure
below shows a net for the box.
15 ft
12 ft
12 ft
17 ft
13.89 ft
13.89 ft
What is the surface area of the box, in square feet, that Zoe
decorates?
Answers
The surface area of the box, in square feet, that Zoe decorates is 430.35 square feet
The surface area of the boxTo determine the surface area from the net of the box, we simply calculate the area of each shape.
From the given figure, we have the following shapes and dimensions:
Rectangles: 15 ft by 12 ft and 13.89 ft by 15 ftTriangle: Base = 12 ft and Height= 7 ftThe area is then calculated using:
Area = Areas of the rectangle + Area of the triangle
So, we have:
Area = 15 * 12 + 13.89 * 15 + 0.5 * 12 * 7
Evaluate the sum of products
Area = 430.35
Hence, the surface area of the box, in square feet, that Zoe decorates is 430.35 square feet
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Answer:577.35 square feet
Step-by-step explanation:
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2
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63
Paula is standing on a bridge that is 110 above
the water below. She throws a pebble over
the edge of the bridge, and it hits the top of a
24 foot high bridge guard. The quadratic
equation that models the path of the pebble is:
p(t) = -16t² + 48t + 110
such that t is the time in seconds and p is
the height of the pebble.
What is the maximum height of the
pebble?
A) 2 feet
(B) 146 feet
C110 feet
D) 210 feet
Answers
Answer:
(B) 146 feet
Step-by-step explanation:
You want to know the maximum height of a pebble whose height is described by p(t) = -16t² + 48t + 110.
Vertex formWe can write the equation in vertex form as follows:
p(t) = -16(t² -3t) +110
p(t) = -16(t² -3t +2.25) +110 +16(2.25)
p(t) = -16(t -1.5)² +146
This shows the vertex of the quadratic curve to be (1.5, 146).
The maximum height of the pebble is 146 feet.
3/4x -9= 27 kinds need help
Answers
Answer:
X= 48
Step-by-step explanation:
math
Way
- It’s a calculator for algebra and stuff
Write the slope-intercept form of the equation of the line perpendicular to the
graph of 2x-y=6 that passes through the point (-4,1).
Answers
Answer:
12x=4y........................................
The following are goals scored by a soccer team at each game in their recent season. 0 0 0 0 0 0 0 0 0 1 |1|11|11| 1 1 2 2 2 2 2 2 23 355 Complete the frequency table.
Answers
The completed frequency table is given by the image presented at the end of the answer.
How to complete the frequency table?The frequency table is completed with the absolute frequencies of each amounts, that is, the number of games in which the given number of goals was scored.
From the first image given at the end of the answer, the distribution of the number of goals of the team is given as follows:
In eight games, the team scored zero goals.In twelve games, the team scored one goal.In three games, the team scored two goals.In one game, the team scored three goals.In zero games, the team scored four goals.In five games, the team scored five goals.Missing InformationThe problem is given by the image at the end of the answer
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Look carefully at the markings on a 10ml graduated cylinder. Referring back to your experience with the rulers, how many decimal places will this measuring device provide
Answers
If the meniscus is on 6. Then the reading will be 6 mL which is 2 decimal places.
What is a cylinder?A cylinder is a closed solid that has two parallel circular bases connected by a curved surface.
Look carefully at the markings on a 10ml graduated cylinder.
Referring back to your experience with the rulers.
Then the number of the decimal places will this measuring device provide will be
Normally the 10-mL graduated cylinders are always read to 2 decimal places.
A. If the bottom of the meniscus is directly on 6, then the reading will be 6.00 mL i.e. 2 decimal places.
B. If the bottom of the meniscus is directly on the third line after 6, the reading will be 6.60 i.e. 2 decimal places.
C. If the bottom of the meniscus is between the third and fourth; line after 6, we can take a middle value between 6.60 and 6.80. Reading could be 6.70. Here also 2 decimal places are used.
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Which of the graphs below shows the solution set for -36 ≤ 2x + 4(x-3)?
A.
B.
-10-9-8-7-6-5-4-3-2-1 0
-10-9-8-7-6-5-4-3-2-1 0
C. A++
▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬
-10-9-8-7-6-5-4-3-2-1 0
D. +++
-10-9-8-7-6-5-4-3-2-1 01
Answers
Considering the given inequality, the solution is given by graph D.
What is the solution to the inequality?The inequality is given by:
-36 ≤ 2x + 4(x-3)
Applying the operations:
-36 ≤ 2x + 4x - 12
-24 ≤ 6x
6x >= -24
x >= -24/6
x >= -4.
Hence option D is correct.
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Random question, BUT is God real?
Answers
Answer:
Yes of course God is real.
Step-by-step explanation:
One way god is to prove that God is real is to simply look at the world in which we live. The beauty, complexity and organization of the material world around us point toward a master Designer.
the amount of food prepared for a catered event depends on the number of people attending. which variable is the explanatory variable? [1 point]
Answers
The number of people attending affects how much food is prepared for a catered event. The number of people attending is the explanatory variable in this scenario.
One kind of independent variable is an explanatory variable. Both words are frequently used interchangeably. This is the variable that changes as we change it or watch it change. In other words, it is a factor that a researcher has changed in an experiment.
In the given situation, the number of people attending is an explanatory variable and the amount of food prepared is a dependent variable. As the number of people attending increases, the amount of food prepared also increases. This means the effect of one variable alters the effect of another variable. The change of one variable is called an independent variable while the change that occurs because of the independent variable is a dependent variable.
Therefore, the required answer is the number of people attending.
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Ana y Maria vieron 2 hombres robar una joyería y también vieron como alcanzaron a salir. cuando fueron interrogadas por la policía, las jóvenes dieron información sobre la placa del auto que vieron alejarse: constaba de dos letras seguidas de cuatro dígitos. María aseguraba que la segunda letra era una O o una Q y que el último dígito era un 3 a un 8. Y Ana dijo que la primera letra era una C o una G y que el primer dígito de la definitivamente un 7. ¿Cuántas placas diferentes tendrá que verificar la policía para dar a conocer el auto que vieron Ana y María?
Answers
have a government
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Step-by-step explanation:
yan napo lahat answer
the magnitude of vector is always negative true or false
Answers
Answer:
False. The magnitude of a vector is always non-negative.
if a1 = 4 and an = -4a n-1 +5 then find value of a5
please help
Answers
Answer:
Step-by-step explanation:
a1 = 4
a2 = -5(4)-1 = -21
a3 =-5(-21)-1 = 104
a4 = -5(104)-1 =-521
a5 =-5(-521)-1 = 2604
Which of the following represent lengths of the sides of a right triangle?
Answers
Answer:
If a triangle is a right triangle, then the lengths of its sides satisfy the Pythagorean Theorem, a2+b2=c2.
Step-by-step explanation:
To determine which choice is correct?
Will mark brainliest
Solve each system by elimination. Show work for answer
4x+9y=5
-4x +7y=11
Answers
Answer:
adding both equations
4x+9y=5
-4x+7y=11
+ + =+
________
0 +16y=16
y=1
putting the value of y in first eqn
4x+9×1=5
4x=5-9
4x=-4
x=-1
so the x is -1 and y is 1
Consider the following linear programming problem: Maximise profit = 2X₁ - X₂ + 2X3 Subject to: 2X₁ + X₂ + 0x3 ≤10 X₁ + 2X₂ - 2X3 ≤ 20 0X₁ + X₂ + 2X3 ≤ 5 X₁ ,X2, X3 > 0 Change the objective function and constraints to simplex format by including the necessary additional variables.
Solve the problem above using the simplex method.
Answers
The linear programming problem is to maximize the profit function, given constraints, using the simplex method.
To convert the problem into the simplex format, we introduce slack variables to transform the inequality constraints into equalities. Let S₁, S₂, and S₃ be the slack variables for the three constraints, respectively. The converted objective function becomes Z = 2X₁ - X₂ + 2X₃ + 0S₁ + 0S₂ + 0S₃. The constraints in the simplex format are:
2X₁ + X₂ + 0X₃ + S₁ = 10,
X₁ + 2X₂ - 2X₃ + S₂ = 20,
0X₁ + X₂ + 2X₃ + S₃ = 5.
Now we can construct the initial simplex tableau:
┌─────────┬───────┬───────┬───────┬───────┬───────┬───────┬───────┐
│ Basis │ X₁ │ X₂ │ X₃ │ S₁ │ S₂ │ S₃ │ RHS │
├─────────┼───────┼───────┼───────┼───────┼───────┼───────┼───────┤
│ Z │ 2 │ -1 │ 2 │ 0 │ 0 │ 0 │ 0 │
│ S₁ │ 2 │ 1 │ 0 │ 1 │ 0 │ 0 │ 10 │
│ S₂ │ 1 │ 2 │ -2 │ 0 │ 1 │ 0 │ 20 │
│ S₃ │ 0 │ 1 │ 2 │ 0 │ 0 │ 1 │ 5 │
└─────────┴───────┴───────┴───────┴───────┴───────┴───────┴───────┘
Using the simplex method, we perform iterations until we obtain the optimal solution. In each iteration, we select the most negative coefficient in the Z row as the pivot column and apply the minimum ratio test to determine the pivot row. The pivot element is chosen as the value where the pivot column and pivot row intersect. We then perform row operations to make the pivot element equal to 1 and all other elements in the pivot column equal to 0.
After performing the necessary iterations, we reach the optimal solution with a maximum profit of 55 units. The values for the decision variables are X₁ = 0, X₂ = 5, and X₃ = 10. The final simplex tableau is:
┌─────────┬───────┬───────┬───────┬───────┬───────┬───────┬───────┐
│ Basis │ X₁ │ X₂ │ X₃ │ S₁ │ S₂ │ S₃ │
RHS │
├─────────┼───────┼───────┼───────┼───────┼───────┼───────┼───────┤
│ Z │ 0 │ 0 │ 1 │ 0.5 │ -1 │ -0.5 │ 55 │
│ X₂ │ 0.5 │ 0 │ 0 │ 0.5 │ -0.5 │ 0 │ 5 │
│ S₂ │ 0.5 │ 1 │ 0 │ -0.5 │ 0.5 │ 0 │ 15 │
│ X₃ │ -0.5 │ 0 │ 1 │ 0.5 │ 0.5 │ -0.5 │ 0 │
└─────────┴───────┴───────┴───────┴───────┴───────┴───────┴───────┘
Therefore, the optimal solution to the linear programming problem is X₁ = 0, X₂ = 5, and X₃ = 10, with a maximum profit of 55 units.
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Here is a data set: 22, 4, 26, 31, 33, 24
.Answer true or false for the following statements.
If you remove the outlier: = 4
- the range will decrease:
- the mean will increase:
- the median will stay the same:
Answers
Answer:
Step-by-step explanation:
Firstly, removing the 4 will cause the median to change.
Secondly,
Range = highest number - lowest number
With the outlier:
Range = 33 - 4 = 29
Without the outlier:
Range = 33 - 22 = 11
The range has decreased.
Removing the outlier will not change the range and median, but it will increase the mean.
Explanation:The range is the difference between the largest and smallest values in a data set. If you remove the outlier, which is 4 in this case, the smallest value will change, but the largest value will remain the same. Therefore, the range will stay the same.
The mean is the average of all the values in a data set. When you remove an outlier, it significantly affects the average because it is an extreme value. In this case, removing 4 as an outlier will increase the mean value.
The median is the middle value in a data set. Since the median is not affected by extreme values, removing the outlier will not change the median. In this case, the median will stay the same.
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If 2 coins can be exchanged for 3.3782 dubloons, how many dubloons can be obtained for 9 coins?
(Round to the nearest hundredth)
Answers
Answer:
5.33
Hope this helps
A system may be found in one of the three states: operating, degraded, or failed. When operating, it fails at the constant rate of 2 per day and becomes degraded at the rate of 3 per day. If degraded, its failure rate increases 5 per day. Repair occurs only in the failure mode and is to the operating state with a repair rate of 7 per day. If the operating and degraded states are considered the available states, determine the steady- state availability.
Answers
The steady-state availability of the system is 0.625.
The steady-state availability of a system refers to the probability that the system is in an operable state when it is being considered for use. In this scenario, the system can exist in one of three states: operating, degraded, or failed. To determine the steady-state availability, we need to calculate the probability of the system being in the operating state.
Let's denote the probability of the system being in the operating state as P(o) and the probability of the system being in the degraded state as P(d). Since there are only two available states (operating and degraded), the probability of the system being in the failed state can be calculated as 1 - P(o) - P(d), as the probabilities of all states must sum up to 1.
When the system is in the operating state, it fails at a constant rate of 2 per day. This means that on average, two failures occur in a day while the system is in operation. Similarly, when the system is in the degraded state, the failure rate increases to 5 per day.
However, repair can only happen in the failure mode and is always directed towards restoring the system to the operating state, with a repair rate of 7 per day.
To calculate P(o), we can set up the following equation based on the principle of steady-state availability:
P(o) = (repair rate) / (repair rate + failure rate in operating state)P(o) = 7 / (7 + 2)P(o) = 7 / 9P(o) = 0.7778Therefore, the steady-state availability of the system, which represents the probability of it being in an operable state, is 0.7778 or approximately 0.778.
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8-2. Skills Practice. Multiplying a Polynomial by a Monomial.
Answers
To multiply a polynomial by a monomial, you must first identify the terms in the polynomial, as well as the factors in the monomial. Once you have identified the components, use the distributive property to multiply each term of the polynomial by the monomial. Then, combine like terms and simplify the resulting polynomial.
For example:
To multiply 3x2 + 2x - 5 by 2x, you would use the distributive property to multiply each term of the polynomial by the monomial: 3x2 * 2x = 6x3, 2x * 2x = 4x2, and -5 * 2x = -10x. The resulting polynomial is 6x3 + 4x2 - 10x.Learn more about Polynomial: https://brainly.com/question/24662212
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If a Markov chain has the following transition matrix, then what are the long-term probabilities for each state?
[0.6 0.3 0.1]
[0.1 0.8 0.1]
[0.6 0 0.4]
Enter extact answer.
Answers
The long-term probabilities for state 1, state 2, and state 3 are 0.471, 0.314, and 0.216, respectively.
To find the long-term probabilities for each state, we need to find the steady-state vector (also called the equilibrium distribution or limiting distribution) of the Markov chain. This can be done by solving the system of equations:
πP = π
where π is the row vector of long-term probabilities and P is the transition matrix.
We can write this system of equations as:
0.6π1 + 0.1π2 + 0.6π3 = π1
0.3π1 + 0.8π2 + 0π3 = π2
0.1π1 + 0.1π2 + 0.4π3 = π3
We also know that the sum of the probabilities in the long-term must be 1:
π1 + π2 + π3 = 1
We can solve this system of equations to find the long-term probabilities:
π1 = 0.471
π2 = 0.314
π3 = 0.216
Therefore, the long-term probabilities for state 1, state 2, and state 3 are 0.471, 0.314, and 0.216, respectively.
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Sarah started with 40 pieces of gum and gave away x pieces . Gerardo started with 24 pieces of gum and gave away twice as many pieces Sarah did . how many pieces of gum did Sarah give away if they had same number of pieces of gum left
Answers
HELP
According to the Rational Root Theorem, Negative 2/5 is a potential rational root of which function?
f(x) = 4x4 – 7x2 + x + 25
f(x) = 9x4 – 7x2 + x + 10
f(x) = 10x4 – 7x2 + x + 9
f(x) = 25x4 – 7x2 + x + 4
Answers
Answer: Answer is f(x) = 25x4 - 7x2 + x + 4
Step-by-step explanation:
Answer:
d on edg
Step-by-step explanation:
when thuong real estate sold kate's house, she was charged in commission of $4,000. If the house sold for $80,000, what percent commission did the real estate company charge?
Answers
Step-by-step explanation:
when thuong real estate sold kate's house, she was charged in commission of $4,000. If the house sold for $80,000, what percent commission did the real estate company charge?
4,000 / 80,000 = 0.05
0.05 = 5% commission
The proof that UX ≅ SV is shown.
Given: △STU an equilateral triangle
∠TXU ≅ ∠TVS
Prove: UX ≅ SV
Triangle T X V is shown. Point S is on side T X and point U is on side T V. A line is drawn from points S to U to form equilateral triangle T S U. Lines are drawn from point S to point V and from point U to point X and intersect at point W.
What is the missing statement in the proof?
Statement
Reason
1. ∠TXU ≅ ∠TVS 1. given
2. ∠STV ≅ ∠UTX 2. reflex. prop.
3. △STU is an equilateral triangle 3. given
4. ST ≅ UT 4. sides of an equilat. △ are ≅
5. ? 5. AAS
6. UX ≅ SV 6. CPCTC
△SXU ≅ △TVS
△UVX ≅ △SXV
△SWX ≅ △UWV
△TUX ≅ △TSV
Answers
Answer:
making it very simple it is D Edge 2021
Step-by-step explanation:
The correct option is △ TUX ≅ △ TSV
What is CPCTC?CPCTC stands for "corresponding parts of congruent triangles are congruent" and tells us if two or more triangles are congruent, then their corresponding angles and sides are congruent as well.
Given that, △STU an equilateral triangle and ∠TXU ≅ ∠ TVS, and UX ≅ SV by CPCTC so,
∠TXU ≅ ∠TVS → given
∠STV ≅ ∠UTX → reflex. prop.
△STU is an equilateral triangle → given
ST ≅ UT → sides of an equilat. △ are ≅
△ TUX ≅ △ TSV by AAS rule
UX ≅ SV by CPCTC
According to the given proof, the only option fitting is option D.
Hence, The correct option is △ TUX ≅ △ TSV
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the measurement of the edge of a cube is found to be 15 inches, with a possible error of 0.03 inch. (a) use differentials to approximate the possible propagated error in computing the volume of the cube. (b) use differentials to approximate the possible propagated error in computing the surface area of the cube.
Answers
The approximated possible propagated error in computing the volume of the cube is 20.25 .
The approximated possible propagated error in computing the surface area of the cube is 5.4.
The formula's for volume and surface area of cube is as follows,
\(V=a^3\\S=6a^2\)
Where a is the side of the cube.
When you utilize those questionable measurements to calculate something else, error propagation (or uncertainty propagation) occurs to measurement mistakes.
The error in the edge of the cube is 0.03 inch, so
\(a=15\pm0.03\)
Δa=0.03 inches
Now, differentiating the volume function of cube,
\(\frac{dV}{da} =3a^2\\dV=3a^2da\)
Putting the values,
Δ\(dV=3(15)^2\times0.03\\dV=20.25\)
Now, differentiating the surface area function of cube,
\(\frac{dS}{da}=12a\)
Putting the values,
\(dS=12\times15\times da\\dS=12\times15\times 0.03\\dS=5.4\)
Therefore, The approximated possible propagated error in computing the volume of the cube is 20.25 and The approximated possible propagated error in computing the surface area of the cube is 5.4.
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a school cafeteria wants to find out what combinations of food and beverage students are ordering. the results are represented in the two-way table below: food pizza hamburger salad water 11 7 23 beverage slurpee 18 19 5 soda 28 12 14 of all the students, what is the probability that a student ordered a hamburger and a slurpee?
Answers
0.5 is the probability that a student ordered a hamburger and a slurpee, in the school cafeteria where students are ordering food and beverage.
Marginal total of hamburger=7+19+12= 38
Total no. of students who ordered hamburger and slurpee= 19
Total 157 students out of which 38 ordered a hamburger.
We can see that 28 of the total students order pizza as well as soda.
The probability that a student also ordered a soda is given as =
Number of Students who ordered hamburger and slurpee/Total number of students who ordered a hamburger
=19/38
=0.5
The probability that a student ordered a hamburger and a slurpee is 0.5.
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x+1.6=3.77
solve for x
Answers
X = 2.17
Answer:
2,17
Step-by-step explanation:
because
1x=3,77-1,6/1
x=2,17