let t(n) be number of all the positive divisors of n. prove that t(n) is odd only if n is a perfect square

a. Original statement: If n is any prime number that is greater than 2, then n+1 is even.

Converse: If n+1 is even, then n is prime and greater than 2.

Counterexample: n = 4, which is not a prime number but 4+1=5 is a prime number greater than 2.

b. Original statement: If m is any odd integer, then 2m is even.

Converse: If 2m is even, then m is odd.

Counterexample: m = 2, which is not an odd integer but 2×2=4 is an even integer.

c. Original statement: If two circles intersect in exactly two points, then they do not have a common center.

Converse: If two circles do not have a common center, then they intersect in exactly two points.

Counterexample: Two identical circles with centers coinciding have a common center and intersect at all points on the circle, which is more than two points.

In mathematics, the converse of a conditional statement is formed by interchanging the hypothesis and conclusion.

For example, if the original statement is "If p then q",

the converse statement is "If q then p".

The converse statement may or may not be true, and it needs to be proven separately.

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The kinetic energy (in joule) of the asteroid is, 33 million

The kinetic energy is the energy associated with the particle having mass m and moving with the velocity v.

To find the kinetic energy of the asteroid, we need to use the formula:

KE = (1/2)mv²

where KE is the kinetic energy, m is the mass of the asteroid, and v is its velocity.

The mass of the asteroid can be calculated using its density and volume:

V = (4/3)πr³

where V is the volume of the asteroid, r is its radius, and π is the mathematical constant pi.

Since the asteroid is spherical, its radius is half its diameter, which is 0.54 km.

So,

V = (4/3)π(0.54 km)³

  = 0.65 km³

The mass of the asteroid is then:

m = density × volume

   = 3000 kg/m³ × 10⁹ m³/km³ × 0.65 km³

   = 1.95 × 10¹² kg

Now we can calculate the kinetic energy:

KE = (1/2)mv²  

     = (1/2) × 1.95 × 10¹² kg × (23.0 km/sec)²

     = 1.32 × 10²³ joule

To convert this to megatons of TNT, we divide by the energy of one megaton:

= 1.32 × 10³ joule / (4 × 10¹⁵ joule/megaton)

= 3.3 × 10⁷ megatons

Hence, the KE of asteroid is 33 million

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Answer: V = 493.19

Step-by-step explanation:

V=1/3bh

7.6*12.4=94

94*15.7=1479.568

1479.568*1/3=493.19

Dylan's father's age would be 36 years old at the time when Dylan is 6 years old.

In mathematics, an expression or mathematical expression is a finite combination of symbols that is well-formed according to rules that depend on the context.

Mathematical symbols can designate numbers (constants), variables, operations, functions, brackets, punctuation, and grouping to help determine order of operations and other aspects of logical syntax.

Given is that Dylan's father's age is currently the square of Dylan's age. Dylan is 6 years old.

We can write Dylan's father's age as -

A{father} = A{Dylan} x A{Dylan}

A{father} = 6 x 6

A{father} = 36

Therefore, Dylan's father's age would be 36 years old at the time when Dylan is 6 years old.

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{Question in english -
Dylan's father's age is currently the square of Dylan's age. If Dylan is 6 years old, how old is his dad?}

Answer:

x^2+8x+15

x^2-2x+15

x^2-8x+15

2x^2&10x-28

3x+3n+20 is the answer

Answer:

The formula to find the area of a regular pentagon (a polygon with five sides of equal length and equal interior angles) is:

Area = (1/4) * sqrt(5(5+2sqrt(5))) * s^2

Where s is the length of one side of the pentagon.

Step-by-step explanation:

Taking into account the definition of a system of linear equations, the number of pennies and nickles that Arianys has is 15 and 6 respectively.

The degrees of systems of linear equations are groupings of linear equations with the same unknowns, of which it is necessary to find a common solution.

Solving a system of equations consists of finding the value of each unknown so that all the equations of the system are satisfied.

In this case, a system of linear equations must be proposed taking into account that:

You know:

The system of equations to be solved is

x + y = 21

0.01x + 0.05y = 0.45

There are several methods to solve a system of equations, it is decided to solve it using the graphical method, which consists of representing the graphs associated with the equations of the system to deduce its solution. The solution of the system is the point of intersection between the graphs, since the coordinates of said point satisfy both equations.

In this case, graphing the system of equations (image attached) it is obtained that the system of intersection between both equations is (x,y)=(15,6). This means that the number of pennies that Arianys has is 15 and the number of nickels that Arianys has is 6.

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We can start by using the formula for inverse square law: F = k/d^2. where F is the force, d is the distance between the magnets, and k is a constant.

We can use the given information to solve for k: 4 = k/0.06^2, k = 4 * 0.06^2, k = 0.0144

Now we can use the value of k to find the force when the magnets are 0.03 meters and 0.1 meters apart: F1 = 0.0144/0.03^2 = 16, F2 = 0.0144/0.1^2 = 0.144

The work required to move the magnets is equal to the change in potential energy between the initial and final positions.

We can use the work-energy principle, which states that the work done on an object is equal to its change in kinetic energy plus its change in potential energy. Since the magnets are not moving, their kinetic energy is constant, so the work done on them is equal to their change in potential energy: W = Uf - Ui

where W is the work, Uf is the final potential energy, and Ui is the initial potential energy. The potential energy of the magnets is given by: U = -k/d

where k is the constant we found earlier and d is the distance between the magnets.

Therefore, the initial potential energy is: Ui = -0.0144/0.03 = -0.48

And the final potential energy is: Uf = -0.0144/0.1 = -0.144

So the work required to move the magnets is: W = -0.144 - (-0.48) = 0.336 Joules

Therefore, the work required to move the magnets from 0.03 meters apart to 0.1 meters apart is 0.336 Joules (rounded to three decimal places).

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To find out how much interest will be paid on a $155,000 house if a monthly payment of $850 is made for 30 years, we can use the formula for the total interest paid on a fixed-rate mortgage loan:

Total Interest = (Monthly Payment x Number of Payments) - Loan Amount

where Monthly Payment is $850, Number of Payments is 30 years x 12 months/year = 360 months, and Loan Amount is $155,000.

Plugging in these values, we get:

Total Interest = ($850 x 360) - $155,000

Total Interest = $306,000 - $155,000

Total Interest = $151,000

Therefore, the total interest paid on the $155,000 house over 30 years with a monthly payment of $850 is $151,000.

The areas of the squares are calculated from the Pythagoras relation of right angled triangle and it is 40 ft², 55 ft², 95 ft²

A triangle is a plane figure or polygon with three sides and three angles.

A Triangle has three vertices and the sum of the interior angles add up to 180°

Let the Triangle be ΔABC , such that

∠A + ∠B + ∠C = 180°

The area of the triangle = ( 1/2 ) x Length x Base

For a right angle triangle

From the Pythagoras Theorem , The hypotenuse² = base² + height²

if a² + b² = c² , it is a right triangle

if a² + b² < c² , it is an obtuse triangle

if a² + b² > c² , it is an acute triangle

Given data ,

Let the right angle triangle be represented as ΔABC

Now , the measure of side AB = √40 feet

The measure of side BC = √55 feet

The measure of side AC = √95 feet

For a right angle triangle

From the Pythagoras Theorem , The hypotenuse² = base² + height²

Substituting the values in the equation , we get

AC² = AB² + BC²

( √95 )² = ( √40 )² + ( √55 )²

On simplifying the equation , we get

95 = 40 + 55

95 = 95

Therefore , the area of squares formed is 40 ft², 55 ft², 95 ft²

Hence , the triangle is having sides √40 feet , √55 feet and √95 feet

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The equation of line is y = 5x + 3 , where the slope is m = 5

The equation of a line is expressed as y = mx + b where m is the slope and b is the y-intercept

And y - y₁ = m ( x - x₁ )

y = y-coordinate of second point

y₁ = y-coordinate of point one

m = slope

x = x-coordinate of second point

x₁ = x-coordinate of point one

The slope m = ( y₂ - y₁ ) / ( x₂ - x₁ )

Given data ,

Let the equation of line be represented as A

Now , the value of A is

Let the first point be P ( 0 , 3 )

Let the second point be Q ( 5 , 28 )

Now , the slope of the line is m = ( y₂ - y₁ ) / ( x₂ - x₁ )

Substituting the values in the equation , we get

Slope m = ( 28 - 3 ) / ( 5 - 0 )

Slope m = 25 / 5 = 5

Now , the equation of line is y - y₁ = m ( x - x₁ )

Substituting the values in the equation , we get

y - 3 = 5 ( x - 0 )

On simplifying the equation , we get

y - 3 = 5x

Adding 3 on both sides of the equation ,  we get

y = 5x + 3

Hence , the equation of line is y = 5x + 3

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Both measurements are below the minimum area of 169 square feet required.

Area is a measurement of size of two dimensional surface such as Square rectangle or circle. It is calculated by multiplying the length of the surface by width area also can measured in terms of square units such as square feet or square meters area is an important concept in mathematics and is used to measure the size of safe and object it is also used to find the total area of a group of shape.

The measurements of the parking spaces shown do not meet the requirements of the town. The first parking space has an area of 144 square feet, while the second parking space has an area of 128 square feet. Both measurements are below the minimum area of 169 square feet required.

The area of the figure can be found by using the formula for the area of a regular polygon. Since the figure is composed of four 60-degree angles and four sides, the formula can be expressed as:

A = (1/2) * (a * s) * n

Where a is the length of one side of the polygon, s is the length of the apothem (a line perpendicular to the center of the polygon to a corner), and n is the number of sides of the polygon.

Since the figure has four sides of equal length (5 meters each), the length of the apothem can be calculated by using the Pythagorean theorem. The equation for the apothem is:

s = [tex](4^{2} - 5^{2} ) ^{0.5}[/tex]

Plugging these values into the area formula yields:

A = (1/2) * (5 * 4.472) * 4

A = (1/2) * (22.36) * 4

A = 44.72 square meters

Rounding to the nearest tenth yields an area of 44.7 square meters.

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Assume n is odd. Since n^3 is odd, n^3 + 5 should be even, but it's odd. Therefore, n is even.

The correct ranking of the options is:

As n^3 and 5 are odd, their sum n^3 + 5 should be even, but it is given to be odd. This is a contradiction.

Suppose that n^3 + 5 is odd and that n is odd. We know that the sum of two odd numbers is even. As n is odd, n^3 is odd. Therefore, our supposition was wrong; hence n is even.

The proof by contradiction shows that the assumption that n is odd leads to a contradiction with the given fact that n^3 + 5 is odd. Therefore, the assumption that n is odd must be false, and hence n is even.

The proof begins by assuming that n is odd and showing that this leads to a contradiction with the given fact that n^3 + 5 is odd. Specifically, since the sum of two odd numbers is even, n^3 + 5 should be even if n is odd. However, we know that n^3 + 5 is odd, so our assumption that n is odd must be false, meaning that n is even.

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2 - 2i is the quotient of these complex numbers (6-7i)-(4-5i).

A division is a process of splitting a specific amount into equal parts.

To find the quotient of complex numbers, we need to use the formula:

(a + bi) / (c + di) = [(a + bi) x (c - di)] / (c^2 + d^2)

where a, b, c, and d are real numbers and i is the imaginary unit.

In this case, we are subtracting two complex numbers:

(6-7i) - (4-5i)

= 6 - 7i - 4 + 5i (distribute the negative sign)

= 2 - 2i

Hence,  2 - 2i is the quotient of these complex numbers (6-7i)-(4-5i).

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According to the figure of the right triangle

When a right triangle has one of the angles as 45 degrees the angle must be 45 degrees two. Also, the sides are equal, hence we have that

a = b

Using trigonometry we solve for b as follows

cos 45 = b / 10√2

multiplying both sides by 10√2

10√2 * cos 45 = 10√2 * b / 10√2

10√2 * cos 45 = b

rearranging

b = 10√2 * cos 45

where cos 45 = 1/√2 = √2/2

b = 10√2 * 1/√2

b = 10

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A Reasonable prediction for the number of times a white or red marble will be drawn is 3/10 x 180 = 54. Thus, option b is correct

If a marble is drawn and not replaced, the probability of drawing an orange marble on the second draw depends on the color of the marble drawn on the first draw. If an orange marble was drawn on the first draw, then there will be one less orange marble in the bag, and the probability of drawing another orange marble on the second draw will be 4 out of the remaining 29 marbles. If a non-orange marble was drawn on the first draw, then there will be 5 orange marbles in the bag, and the probability of drawing an orange marble on the second draw will be 5 out of 29.

There are a total of 30 marbles in the bag.

The probability of drawing a white or red marble is

2/30 + 7/30

= 9/30

= 3/10.

Therefore, a reasonable prediction for the number of times a white or red marble will be drawn is 3/10 x 180 = 54. Answer B.

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Work Shown:

5r + p

5*4 + 3

20 + 3

23

The worst month for the team in terms of results was October, with a total of three losses in five matches.

Number is a mathematical entity used to represent a computer magnitude it can be symbol or a combination of simple you should in order to take quantity such as the two, five, seven, three or eight number are used to verify the context including counting measuring and computing.


The team won only one match and drew one, resulting in a total of five points. This was the lowest points total of any month during the season, a stark contrast to the other months where the team achieved much higher totals. The team's form during October was a far cry from their form during the rest of the season, with only one win in five matches. This poor form contributed to the team's overall lack of success during the season and led to their eventual relegation from the top division.

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The value of x and y are 13 and 3 respectively and the measure of angle mqr  is 51 degree.

It is an measure of rotation between one extended side(we extend it virtually) of the polygon with its adjacent side which is not extended.

WE are given that in the diagram, Ks, LT, MU are parallel to each other.

Angle knj = (2x-5)°, Angle lpn = (2y + 15), Angle pqu = (3x+4y)°.

Sine, the corresponding angles are equal

(2x-5) = (2y+15)

2x - 2y = 15 + 5

2x - 2y = 20

x - y = 10

x = 10 + y

Now,

(3x+4y) = (2y+15)

3x + 4y - 2y = 15

3x + 2y = 15

3(10 + y) + 2y = 15

30 + 5y = 15

y = 3

x = 13

Angle mqr = (3x+4y) = 51 degree.

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By solving an expression, we can find that the number of packs of paper that each classroom should get is 10/3.

Expressions are mathematical statements that comprise either numbers, variables, or both and at least two terms associated by an operator. Addition, subtraction, multiplication, and division are examples of mathematical operations. An example is the expression x + y, which combines the terms x and y with an addition operator. In mathematics, there are two different types of expressions: algebraic expressions, which also include variables, and numerical expressions, which solely comprise numbers.

Given,

Number of classrooms= 3

Number of packs of paper= 10

Now let the number of packs of paper per class be x.

x= Number of packs of paper/Number of classrooms

 = 10/3.

Therefore, each classroom will get 10/3 packs of paper.

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The mean and median of given data of population change for the ten most populous counties in the US from 2000 to 2010 is 7.2% and 6.25%.

To find the mean of the data, we need to add up all the percent changes and divide by the total number of counties:

Mean = (3.1 - 3.4 + 20.3 + 24.2 + 10.0 + 5.8 + 1.6 + 10.8 + 6.7 + 0.1) / 10 = 7.2%

Therefore, the mean percent change for the ten most populous counties in the US from 2000 to 2010 is 7.2%.

To find the median of the data, we first need to order the percent changes from smallest to largest:

-3.4, 0.1, 1.6, 3.1, 5.8, 6.7, 10.0, 10.8, 20.3, 24.2

Since there are an even number of values, the median is the average of the two middle values, which in this case are 5.8 and 6.7. Therefore, the median percent change for the ten most populous counties in the US from 2000 to 2010 is (5.8 + 6.7) / 2 = 6.25%.

The better description of the center of this data is the median, because it is less affected by outliers than the mean. In this case, the percent change for Maricopa County, AZ (24.2%) is much larger than the other values, and it pulls the mean upward. The median, on the other hand, is not affected by outliers, and it gives a better representation of the typical percent change for these counties.

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Answer:

x = 3, y = -1

Step-by-step explanation:

4x - 3y = 15

x + y = 2 -> y = 2 - x

4x - 3(2 - x) = 15

4x - 6 + 3x = 15

7x = 21

x = 3

y = 2 - x

y = 2 -3

y = -1

Answer: x =3, y = -1

Step-by-step explanation:

[tex]\bf{\underline{We\:solve\:by\:applying\:the\:reduction\:method.}}[/tex]

          [tex]\boxed{\large\displaystyle\text{$\begin{gathered}\sf \bf{The\:exercise\:is \ ---\to \ \left \{ {{4x-3y=15} \atop {x+y=2 \ \ \ \ \ }} \right. } \end{gathered}$}}[/tex]

Multiply the second equation by -4, then add both equations.

  4x - 3y = 15

  -4(x + y = 2)

We add these equations to eliminate x.

    -7y = 7

Then we solve -7y = 7 for y. (We divide by 7)

        [tex]\bf{\dfrac{-7y}{ -7}=\dfrac{7}{-7} } \\ \\ \bf{y=-1}[/tex]

We place the found value of y , in one of the original equations y in order to solve for x:

        4x - 3y = 15

        4x - 3(-1) = 15

        3x + 4 = 15

We add (-3) to both sides.

     4x + 3 + (-3) = 15 + (-3)

      4x = 12

We divide both sides by 4.

         [tex]\bf{\dfrac{4x}{4}=\dfrac{12}{4} } \\ \\ \bf{x=3}[/tex]

Solution: x=3,y=-1

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Answer: [tex]x=3.5[/tex]

Step-by-step explanation:

Let [tex]x[/tex] be the number, then

[tex]\frac{1}{2}x=x-7\\\frac{1}{2}x-x=-7\\-\frac{1}{2}x=-7\\x=3.5[/tex]

Simplifying the equations will give 3x -1 = 4x, 3x + 18 = 4x + 7 and 7x = 1

Lets simplify each of the equations in order to match with the  equivalent equation from the second set

3x+6=4x+7

Collecting like terms

3x +6 - 7 = 4x

3x -1 = 4x

The second equation

3(x+6)=4x+7

expanding the bracket

3x + 18 = 4x + 7

The third equation

4x+3x=7-6

simplifying the equation

7x = 1

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The discrete probability distribution is given as follows:

A probability is calculated as the division of the desired number of outcomes by the total number of outcomes.

There are 153 nests, hence the distribution in this problem is constructed dividing each of the amounts by 153, as follows:

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When comparing exponential functions, the one with the bigger base expands more quickly. A logarithmic function grows more quickly if its argument is bigger.

Compared to polynomial functions, which in turn expand more quickly than logarithmic functions, exponential functions grow more quickly.

When comparing exponential functions, the one with the bigger base expands more quickly.

A logarithmic function grows more quickly if its argument is bigger.

We can arrange the provided functions in this manner by applying these rules:

log n, n, n log n, n, n2, n!, nn

Taking the function with the fastest rate of growth first: log n n n log n n 2

Each of these variables has a polynomial or logarithmic growth rate. The exponential functions will now be discussed:

2^n > n^2

Since polynomial functions increase more slowly than exponential functions, n2 grows more slowly than 2n.

The two categories with the fastest growth are: n! > n^2

n! grows more quickly than 2n and n2, as factorial functions expand even more quickly than exponential functions do. Finally:

n^n > n!

Therefore, nn grows more quickly than n! because exponential functions with bigger bases scale up more quickly than those with smaller bases.

Consequently, the following functions are listed in ascending order of growth rate:

log n n n log n n n log n n n! n n.

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The complete statement is 14 tires is 5.6% of 250 tires.

From the question, we have the following parameters that can be used in our computation:

14 tires is% of 250 tires.

As an equation, we have

14 = x% * 250

Divide both sides of the equation by 250

So, we have the following representation

x% = 14/250

Evaluate

x% = 0.056

Multiply by 100

x = 5.6

Hence, the expression in the blank is 5.6

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The average rate of change in the number of Americans over the age of 65 from 2000 to 2015 is approximately -2.09 million people per year.

To find the average rate of change in the number of Americans over the age of 65 from 2000 to 2015, we need to calculate the change in the number of people over 65 during that time period and divide it by the number of years:

Number of people over 65 in 2015:

n(15) = 0.00246(15)^2 + 0.118(15) + 0.183

n(15) = 2.781 million people

Change in the number of people over 65 from 2000 to 2015:

2.781 million - 34.42 million = -31.639 million people

Average rate of change:

-31.639 million / 15 years = -2.09 million people per year

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