To graph the function [tex]y = x^3 - 3x^2 - 144x - 140[/tex], we can analyze its key features using calculus techniques. A clear, labeled sketch of the function will provide a visual representation of these features.
a. To find the x-intercepts, we set y = 0 and solve for x. In this case, we can factor the equation as (x+1)(x+10)(x-14) = 0, so the x-intercepts are x = -1, x = -10, and x = 14. The y-intercept occurs when x = 0, so [tex]y = 0^3 - 3(0)^2 - 144(0) - 140 = -140[/tex].
b. To find local max/min coordinates, we take the derivative of the function and set it equal to zero. The derivative of [tex]y = x^3 - 3x^2 - 144x - 140[/tex] is [tex]y' = 3x^2 - 6x - 144[/tex]. Solving [tex]3x^2 - 6x - 144 = 0[/tex] gives x = 8 and x = -6. We can then evaluate the function at these x-values to find the corresponding y-values.
c. To determine intervals of increasing or decreasing, we analyze the sign of the derivative. When y' > 0, the function is increasing, and when y' < 0, the function is decreasing. We can use the critical points found in part b to determine the intervals.
d. Points of inflection occur when the concavity changes. To find them, we take the second derivative of the function and set it equal to zero. The second derivative of [tex]y = x^3 - 3x^2 - 144x - 140[/tex] is y'' = 6x - 6. Setting 6x - 6 = 0 gives x = 1, which represents the point of inflection.
e. To determine intervals of concavity, we analyze the sign of the second derivative. When y'' > 0, the function is concave up, and when y'' < 0, the function is concave down. We can use the point of inflection found in part d to determine the intervals.
By considering these key features and plotting the corresponding points, we can sketch the function y = x^3 - 3x^2 - 144x - 140 on a grid, ensuring all the identified features are labeled and clear.
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Please just give me the answer
9514 1404 393
Answer:
8
Step-by-step explanation:
The Pythagorean theorem tells you of the relation ...
x² + 6² = 10² . . . . . . . . . squares of sides total to the square of hypotenuse
x² = 100 -36 = 64 . . . . . subtract 6²
x = √64 = 8 . . . . . . . . . . square root
The length of side x is 8 units.
You are charged $16. 05 after tax for a meal. Assume sales tax is 7%, what was the menu price for the meal
Answer:
$15
Step-by-step explanation:
107% is the price of $16.05
so, 107% = $16.05
Divide both sides by 107:
1% = $0.15
Multiply both sides by 100:
100% = $15.00
Will mark brainliest for the **CORRECT** answer!
Answer:
4x + 12x = 320
16x = 320
x = 20
Step-by-step explanation:
This is because the diagram shows 4x + 12x and the total being, 320.
4x + 12x = 16x
and 320/16 = 20
so x = 20
hope this helped :)
For 2 + 5(x-3) > 3x + 11, the answer is x>
Answer:
x >12
Step-by-step explanation:
2 + 5(x-3) > 3x + 11
Distribute
2 + 5x - 15 > 3x+11
Combine like terms
5x -13 > 3x+11
Subtract 3x from each side
5x-13-3x> 3x+11-3x
2x-13 > 11
Add 13 to each side
3x-13+13> 11+13
2x> 24
Divide by 2
2x/2 > 24/2
x >12
Del enunciado: " De cada 2 conejos, hay 5 gallinas" ¿cuál es la razón entre gallinas y total de animales? *
Answer:
La razón entre el número de gallinas y el total de animales es: [tex]\frac{5}{7}[/tex]
Step-by-step explanation:
La razón es una comparación entre dos magnitudes comparables.
En otras palabras, la razón es el cociente entre dos números o dos cantidades comparables entre sí, expresado como fracción.
En este caso, la cantidad total de animales es la suma de la cantidad de conejos y la cantidad de gallinas:
2 conejos + 5 gallinas= 7 animales
Entonces la razón entre el número de gallinas y el total de animales es: [tex]\frac{5}{7}[/tex]
Please answer if your know
Answer:
52 pounds
Step-by-step explanation:
52 pounds
Diane has $334 in her checking account. She writes a check for $112, makes a deposit of $100, and then writes another check for $98. Find the amount left in her account.
Select one:
a. $444
b. $214
c. $224
d. $86
Answer:
334 - 112 + 100 - 98
Step-by-step explanation:
in HIJ the measure of J=90 feet, JH=81 feet, HI=9 feet. Find the measure of I to the nearest degree. (Please answer how you got this answer)
Answer:
63°
Step-by-step explanation:
1) Examing that right triangle, we can find the value of the angle x starting by using a trig ratio and then its reciprocal. Like this:
sin( x ) = 81/91
x = arcsin (81/91)
x = 62.88 = 63
Notice we have the opposite leg and the hypotenuse, so we started with the sine. And to find the value of the angle the arcsin.
2) So, that angle x is 63° (rounding off to the nearest degree)
An article suggested that yield strength (ksi) for A36 grade steel is normally distributed with μ = 42 and σ = 5.0.
(a) What is the probability that yield strength is at most 39? Greater than 60? (Round your answers to four decimal places.)
at most 39 _________. greater than 60 _________. (b) What yield strength value separates the strongest 75% from the others? (Round your answer to three decimal places.)
_______ksi
A)The probability that the yield strength is greater than 60 is approximately 0.0003.
B)The yield strength value that separates the strongest 75% from the others is approximately 45.3725 ksi.
What is probability?
Probability is a fundamental concept in mathematics and statistics that quantifies the likelihood or chance of an event occurring. It provides a numerical measure of uncertainty or the relative frequency with which an event is expected to happen. In simpler terms, probability is a way of expressing how likely it is for a particular outcome or event to take place.
(a) The probability that yield strength is at most 39:
Using the standard normal distribution, we can calculate the z-score as follows:
[tex]\[ z = \frac{{39 - 42}}{{5.0}} = -0.6 \][/tex]
The cumulative probability associated with a z-score of -0.6 represents the probability of obtaining a value less than or equal to 39. Using a standard normal distribution table or a calculator, we find that this cumulative probability is approximately 0.2743.
Therefore, the probability that the yield strength is at most 39 is approximately 0.2743.
The probability that yield strength is greater than 60:
Converting 60 to a z-score:
[tex]\[ z = \frac{{60 - 42}}{{5.0}} = 3.6 \][/tex]
The cumulative probability associated with a z-score of 3.6 represents the probability of obtaining a value greater than 60. Using a standard normal distribution table or a calculator, we find that this cumulative probability is approximately 0.9997.
Since we want the probability of a value greater than 60, we subtract this cumulative probability from 1:
[tex]\[ P(\text{{yield strength}} > 60) = 1 - 0.9997 = 0.0003 \][/tex]
Therefore, the probability that the yield strength is greater than 60 is approximately 0.0003.
(b) The yield strength value that separates the strongest 75% from the others:
To find the yield strength value that separates the strongest 75% from the others, we need to find the z-score corresponding to the cumulative probability of 0.75. Using a standard normal distribution table or a calculator, we find that the z-score associated with a cumulative probability of 0.75 is approximately 0.6745.
Next, we can use the z-score formula to find the yield strength value:
[tex]\[ z = \frac{{x - \mu}}{{\sigma}} \][/tex]
Rearranging the formula to solve for x:
[tex]\[ x = \mu + (z \times \sigma) \][/tex]
Substituting the values into the formula:
[tex]\[ x = 42 + (0.6745 \times 5.0) = 45.3725 \][/tex]
Therefore, the yield strength value that separates the strongest 75% from the others is approximately 45.3725 ksi.
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If y varies inversely with x, and y= 12 when x = 16, what is the constant of variation k?
Answer:
k = 192
Step-by-step explanation:
Given that,
y varies inversely with x. It can be written as :
[tex]y=\dfrac{k}{x}[/tex]
Where
k is the constant of variation
Put x = 16 and y = 12 in the above formula.
[tex]k=yx\\\\k=16\times 12\\\\k=192[/tex]
So, the value of the constant of variation is equal to 192.
) Which relation is a function?
A) y2- x = 8
B) y2 + 3xy = 9y
C) y2 + x = 8x - 8
D) y =3/y- x2
Answer:
It is A
Step-by-step explanation:
If the y2 get subtracted by the 8 then itll be 8
A polygon has the following coordinates: A(-5,2), B(-2,-2), C(2,3), D(6,3), E(6,-5), F(-5,-5). Find the length of EF.
A.
12 units
B.
9 units
C.
11 units
D.
10 units
Answer:
C
Step-by-step explanation:
got it right on edg
How many solutions does the system have?
You can use the interactive graph below to find the answer.
\begin{cases} x+2y=2 \\\\ 2x+4y=-8 \end{cases}
⎩
⎪
⎪
⎨
⎪
⎪
⎧
x+2y=2
2x+4y=−8
Choose 1 answer:
Choose 1 answer:
(Choice A)
A
Exactly one solution
(Choice B)
B
No solutions
(Choice C)
C
Infinitely many solutions
How many solutions does the system have? ⎧
x+2y=2
2x+4y=−8
Answer:
No solution
Step-by-step explanation:
The easiest approach here is to divide the second equation by 2: x + 2y = -4.
Comparing the first equation with this result, we see that the the two lines never intersect, and thus that there is no solution.
At a particular restaurant, each mini hotdog has 100 calories and each slider has 200 calories. A combination meal with mini hotdogs and sliders is shown to have 1200 total calories and 4 times as many mini hotdogs as there are sliders. Graphically solve a system of equations in order to determine the number of mini hotdogs in the combination meal, x,x, and the number of sliders in the combination meal, yy.
Answer:
The number of sliders is 2 and the number of hot dogs is 8.
Step-by-step explanation:
Since at a particular restaurant, each mini hotdog has 100 calories and each slider has 200 calories, and a combination meal with mini hotdogs and sliders is shown to have 1200 total calories and 4 times as many mini hotdogs as there are sliders, in order to determine the number of mini hotdogs in the combination meal, X, and the number of sliders in the combination meal, Y, the following calculation must be performed:
2X + Y = 1200
800 + 400 = 1200
800/100 = 8
400/200 = 2
Thus, the number of sliders is 2 and the number of hot dogs is 8.
6. Brandon and Charlotte Snifflesworth are visiting three of Americas top tourist attractions. The Snifflesworths
live close to Disney World in Orlando, Florida. They decided to start their trip at Disney World before traveling to
San Francisco, California to see the Golden Gate Bridge and then to Minneapolis, Minnesota to go shopping at the
Mall of America. The bearing from Disney to the Golden Gate Bridge is North 65° West. The bearing from the
Golden Gate Bridge to the Mall of America is North 80° East. The bearing from Disney to the Mall of America is
North 30° West and the distance traveled is 1,320 miles. Find all missing interior angles and distances created from
their triangular trip to America's top tourist attractions. Also find the bearing from the Mall of America to
Disneyland.
Answer:
The distance from Disney to the Golden Gate Bridge is 2,162.56 miles
The distance from Mall of America to The Golden Gate Bridge is 1,320 miles
The interior angle at The Golden Gate Bridge is 35°
The interior angle at Mall of America is 110°
The interior angle at Disney is 35°
The direction from Mall of America to Disney is South 30° East
Step-by-step explanation:
The bearing from Disney to the Golden Gate Bridge = North 65° West
The bearing from the Golden Gate Bridge to the Mall of America = North 80° East
The bearing from Disney to the Mall of America = North 30° West
The distance from Disney to the Mall of America = 1,320 miles
Let 'A', 'B', and 'C' represent the interior angles at Disney, The Golden Gate bridge and Mall of America respectively
From the drawing of the triangular trip, we find that the interior angle at C = The sum angles complementary to the bearings at B and at C
Therefore, the interior angle at C = (90° - 65°) + (90° - 80°) = 35°
The interior angle at B = The bearing of C from B - The bearing of A from B
∴ The interior angle at B = 65° - 30° = 35°
B = 35°
∠C = 35° and ∠B = 35°, therefore, the triangle is an isosceles triangle
The interior angle at A = 180° - (∠B + ∠A) = 180° - (35° + 35°) =110°
CA = AB = 1.320
By sine rule, a = sin(110) × 1320/sin(35) ≈ 2,162.56 miles
a = CB = 2,162.56 miles
Therefore, we have;
The distance from Disney to the Golden Gate Bridge = 2,162.56 miles
The distance from Mall of America to The Golden Gate Bridge = 1,320 miles
The interior angle at The Golden Gate Bridge = 35°
The interior angle at Mall of America = 110°
The interior angle at Disney = 35°
The magnitude of the bearing of Mall of America to Disney = The magnitude of the alternate angle to the bearing of Disney to Mall of America = 30°
∴ The direction from Mall of America to Disney = South 30° East
Question 3 [20 marks] Consider two utility functions u(x) and ˜u(x) where x is the amount of money consumed by the agent.
a) Explain formally what it means that an agent with utility function u is more risk averse than an agent with utility function ˜u.
b) Show that an agent with utility function u(x) = log x is more risk averse than an agent with utility function ˜u(x) = √ x.
a) Formal explanation of risk aversion An agent with utility function u is more risk averse than an agent with utility function ˜u if the former has a higher marginal utility of consumption and a diminishing marginal utility of consumption.
The marginal utility of consumption is defined as the amount of utility gained from an additional unit of consumption.
b) Show that an agent with utility function u(x) = log x is more risk averse than an agent with utility function ˜u(x) = √ x. An agent with utility function u(x) = log x is more risk averse than an agent with utility function ˜u(x) = √ x. To show this, we need to find the Arrow-Pratt coefficient of risk aversion, also known as the coefficient of relative risk aversion. The Arrow-Pratt coefficient of risk aversion is given by :-u''(x)/u'(x)Where u'(x) is the first derivative of u with respect to x and u''(x) is the second derivative of u with respect to x.
The Arrow-Pratt coefficient of risk aversion measures the curvature of the utility function. A higher value of the Arrow-Pratt coefficient of risk aversion indicates greater risk aversion. Let us calculate the Arrow-Pratt coefficient of risk aversion for both functions:-For u(x) = log x, u'(x) = 1/x, and u''(x) = -1/x². Therefore, the Arrow-Pratt coefficient of risk aversion for u(x) is given by:-u''(x)/u'(x) = -1/x² ÷ (1/x) = -x For ˜u(x) = √ x, ˜u'(x) = 1/2√ x, and ˜u''(x) = -1/4x^(3/2). Therefore, the Arrow-Pratt coefficient of risk aversion for ˜u(x) is given by:-˜u''(x)/˜u'(x) = -1/4x^(3/2) ÷ (1/2√ x) = -1/2√ x
Therefore, since -x < -1/2√ x, the agent with the utility function u(x) = log x is more risk averse than the agent with the utility function ˜u(x) = √ x.
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Select all that apply. Which numbers are not perfect squares? 36 14 20 16 25 18 24
Answer:
20, 18, 14, and 24 are all not perfect squares
Step-by-step explanation:
Which table represents a quadratic function? No downloadable links or files, I will mark brainliest.
It is not the 3rd option.
Answer:
where is the table lol
Will mark brainliest !!!
Answer:
27
Step-by-step explanation:
Length * width * height to find the volume so it is 3*3*3=27
The sixth grade art students are making a mosaic using tiles in the shape of triangles.Each tile Hans leg measures of 7cm and 4cm. If there are 84 tiles in the mosaic,what I sent the area of the mosaic
Answer:
1,176 square centimeters
Step-by-step explanation:
The computation of the area of the mosaic is shown below:
As we know that
The Area of the triangle is
= 1 ÷ 2 × base × height
= 1 ÷ 2 × 7 × 4
= 14
Now 1 tile would be 14 square centimeters
And, there are 84 tiles in the mosaic
So, the total area is
= 84 × 14
= 1,176 square centimeters
how do you determine if a relation is a function
Answer:
Identify the output values. If each input value leads to only one output value, classify the relationship as a function. If any input value leads to two or more outputs, do not classify the relationship as a function.
Step-by-step explanation:
Tell whether the ordered pair is of liner equations (5,-6) 6x+3y=12 4x+y=14
Answer:
The ordered pair is one of the solution to the system of equations. See below.
Step-by-step explanation:
To tell if the ordered pair is the solution to the system of equations or not, we can do by substituting the ordered pair in both equations.
(x, y) = (5,-6)
Substitute x = 5 and y = -6 in both equations.
First Equation
6x+3y=12
6(5)+3(-6)=12
30-18=12
12=12
Second Equation
4x+y=14
4(5)-6=14
20-6=14
14=14
Because both equations have same sides which mean that both equations are true for (5,-6). Therefore (5,-6) is part of the equations.
solve the system of differential equations. = 2x 3y 1 = -x - 2y 4
The given system of differential equations is:
dx/dt = 2x + 3y
dy/dt = -x - 2y + 4
To solve this system, we can use various methods such as substitution, elimination, or matrix methods. Let's use the matrix method.
First, we can rewrite the system in matrix form:
d/dt [x y] = [2 3] [x] + [1]
[-1 -2] [y] + [4]
Next, we define A as the coefficient matrix [2 3; -1 -2], X as the column matrix [x; y], and B as the column matrix [1; 4]. The system can now be written as:
dX/dt = AX + B
To find the solution, we can calculate the eigenvalues and eigenvectors of matrix A. From the eigenvalues, we determine the corresponding eigenvectors and use them to construct the general solution. However, without the specific values of matrix A, it is not possible to provide the exact solution.
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Find the general solution of the following using determent coefficients. y" - 4y' + 5y = 16 cos (1)
The general solution of the differential equation y" - 4y' + 5y = 16 cos (1) using determinant coefficients is given by y =
yh + yp = c1 e^(2x) cos(x) + c2 e^(2x) sin(x) + 16/((5^2 + 1)√26) cos(1).
In order to find the solution using determinant coefficients, first, we solve the homogeneous equation y" - 4y' + 5y = 0. The characteristic equation is given by r^2 - 4r + 5 = 0, which has roots r = 2 ± i. Therefore, the general solution of the homogeneous equation is yh = c1 e^(2x) cos(x) + c2 e^(2x) sin(x).
Next, we find the particular solution of the non-homogeneous equation using the method of undetermined coefficients. Since the forcing function is cos(1), we assume the particular solution to be of the form yp = a cos(1). Substituting this into the differential equation, we get -a + 4a + 5a cos(1) = 0, which implies a = 16/(5^2 + 1). Hence, the particular solution is yp = 16/((5^2 + 1)√26) cos(1).
Therefore, the general solution of the given differential equation is y = yh + yp = c1 e^(2x) cos(x) + c2 e^(2x) sin(x) + 16/((5^2 + 1)√26) cos(1).
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The distance between bases on a baseball field is 27.43 meters. Joe has jogged from one base to the next 4.5 times. How far has he jogged?
a. 12.48 meters
b. 40.72 meters
c. 123.43 meters
d. 123.45 meters
Number of meters Joe jogged is 123.43 meters. Therefore, the correct answer is option C.
Given that, the distance between bases on a baseball field is 27.43 meters.
Joe has jogged from one base to the next 4.5 times.
Number of meters Joe jogged = Distance between bases on a baseball field × Number of times Joe jogged
= 27.43 × 4.5
= 123.43 meters
Therefore, the correct answer is option C.
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A submarine began at sea level and descended toward the ocean floor at a rate of −0.015 km per minute. Its final depth was −0.3675 km. Estimate how long it took the submarine to reach its final depth by rounding the dividend and divisor to the nearest hundredth.
Estimate of the quotient:
Answer:
Around 24.5 minutes
Step-by-step explanation:
Answer:
Estimate of the dividend: -0.37
Estimate of the divisor: -0.02
Estimate of the quotient: 18.5
Step-by-step explanation:
I did the test and it was right and I dubble checked it to
5 x 100 = ? for easy points
Answer:
500! Just multiply 5 x 1 then add the remaining 2 zeros :)
Step-by-step explanation:
Thank you!
Answer:
[tex]\huge\boxed{\boxed{\underline{\textsf{\textbf{Answer}}}}}[/tex]
[tex]5 \times 100 \\ = 500[/tex]
꧁❣ ʀᴀɪɴʙᴏᴡˢᵃˡᵗ2²2² ࿐
help please! :)) ill do whatever it is that gets you guys points!!! please help!
Answer:
7.5 in.
Step-by-step explanation:
Answer:
7.5 in
Step-by-step explanation:
using pythagorean theorem
Hyp²= Opp² + Adj ²
where 9= hyp
5= adj
and x = opp
9²=x²+5²
81=x² + 25
collect like terms
81-25=X²
56=X²
since we're looking for X and not X²
we square root both sides
√x²=√56
x=7.483 approximately 7.5
HW: using trigonometric identities, show that the solution of the damped forced oscilla from can be written as: (24) Xlt)=12 Fo/m Sin (wo-w)t sin (wotw)t 7 Wo² - w² 2 2 Hint: ure the identifies for addition and Substraction of angles.
Hence, the required equation is `(24) Xlt)=12 Fo/m Sin (wo-w)t sin (wotw)t 7 Wo² - w² 2 2`.
Given damped forced oscillation equation is,`m d²x/dt² + c dx/dt + kx = Fo sin(wt)`Using trigonometric identities, we can write solution for the given damped forced oscillation equation as,X(t) = Acos(wt + Φ) + Xpwhere Xp = (Fo/k) sin(wt - δ)Let's substitute X(t) in the given equation to get the required equation.```
X(t) = Acos(wt + Φ) + Xp
=> dX(t)/dt = -Awsin(wt + Φ) + (Fo/k)wcos(wt - δ)
=> d²X(t)/dt² = -Aw²cos(wt + Φ) - (Fo/k)w²sin(wt - δ)
```Now, substitute these values in the given damped forced oscillation equation.`md²X(t)/dt² + cdX(t)/dt + kX(t) = Fo sin(wt)`⇒ `m(-Aw²cos(wt + Φ) - (Fo/k)w²sin(wt - δ)) + c(-Awsin(wt + Φ) + (Fo/k)wcos(wt - δ)) + k(Acos(wt + Φ) + (Fo/k)sin(wt - δ)) = Fo sin(wt)`Grouping the terms of sines and cosines, we get⇒ `{-Aw²mcos(wt + Φ) + Awcsin(wt + Φ) + (Fo/k)w²sin(δ) + kAcos(wt + Φ) + (Fo/k)wcos(δ)} = Fo sin(wt) - c(Fo/k)wcos(wt - δ)`Let's solve these equations for `δ` and `A`.```
-Aw²mcos(wt + Φ) + Awcsin(wt + Φ) + kAcos(wt + Φ) = 0 .....................(1)
(Fo/k)w²sin(δ) + (Fo/k)wcos(δ) = Fo sin(wt) - c(Fo/k)wcos(wt - δ) .....(2)
```Squaring and adding both equations, we get,`(Aw)²m + kA² = (Fo/k)²`or `A = Fo/(k² - mω²)^(1/2)`From equation (1), we have,`(Aw)²m + kA² = 0`or `δ = tan⁻¹(Aw/k)`Substitute values of A and δ in equation (2), we get,`Xp = (Fo/k) sin(wt - δ) = Fo/(k² - mω²)^(1/2) sin(wt - tan⁻¹(Aw/k))`Therefore, solution for the given damped forced oscillation equation is,`X(t) = Acos(wt + Φ) + Xp`= `12 Fo/m Sin (wo-w)t sin (wotw)t / (wo² - w²)²`
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show that an integer is divisible by 9 if and only if the sum of its decimal digits is divisible by 9. cheg
It is proved here that an integer is divisible by 9 if and only if the sum of its decimal digits is divisible by 9. This is known as divisibility test for 9.
How to test divisibility for 9?
To show that an integer is divisible by 9 if and only if the sum of its decimal digits is divisible by 9, we can use the concept of congruence.
Let's start by representing an integer as the sum of its decimal digits. Consider an integer n expressed in decimal notation as:
[tex]n = d_k * 10^k + d_(k-1) * 10^(k-1) + ... + d_2 * 10^2 + d_1 * 10^1 + d_0 * 10^0[/tex],
where [tex]d_i[/tex] represents the i-th decimal digit of n, and k is the number of digits in n (k >= 0).
We want to prove that n is divisible by 9 if and only if the sum of its decimal digits, [tex]d_k + d_(k-1) + ... + d_2 + d_1 + d_0[/tex], is divisible by 9.
1. If n is divisible by 9:
Assume n is divisible by 9, which means there exists an integer q such that n = 9q. We can express n as:
[tex]n = (d_k * 10^k + d_(k-1) * 10^(k-1) + ... + d_2 * 10^2 + d_1 * 10^1 + d_0 * 10^0) = 9q[/tex]
Since 10 is congruent to 1 modulo 9 (10 ≡ 1 (mod 9)), we can rewrite the above equation as:
[tex](d_k + d_{(k-1)} + ... + d_2 + d_1 + d_0)[/tex] ≡ [tex]9q (mod\ 9)[/tex].
The left-hand side of the congruence represents the sum of the decimal digits, and the right-hand side is a multiple of 9. Therefore, the sum of the decimal digits is divisible by 9.
2. If the sum of the decimal digits is divisible by 9:
Assume the sum of the decimal digits is divisible by 9, which means there exists an integer p such that [tex](d_k + d_{(k-1)} + ... + d_2 + d_1 + d_0) = 9p.[/tex]
We can express n as:
[tex]n = (d_k * 10^k + d_{(k-1)} * 10^{(k-1)} + ... + d_2 * 10^2 + d_1 * 10^1 + d_0 * 10^0) = (9p + d_k * 10^k + d_{(k-1)} * 10^{(k-1)} + ... + d_2 * 10^2 + d_1 * 10^1 + d_0).[/tex]
Since 10 is congruent to 1 modulo 9 (10 ≡ 1 (mod 9)), we can rewrite the above equation as:
n ≡ [tex](9p + d_k + d_{(k-1)} + ... + d_2 + d_1 + d_0)[/tex] ≡ 0 (mod 9).
This shows that n is congruent to 0 modulo 9, or in other words, n is divisible by 9.
Therefore, we have shown that an integer is divisible by 9 if and only if the sum of its decimal digits is divisible by 9.
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