Yi and Sue play a game. They start with the number $42000$. Yi divides by a prime number, then passes the quotient to Sue. Then Sue divides this quotient by a prime number and passes the result back to Yi, and they continue taking turns in this way.

For example, Yi could start by dividing $42000$ by $3$. In this case, he would pass Sue the number $14000$. Then Sue could divide by $7$ and pass Yi the number $2000$, and so on.

The players are not allowed to produce a quotient that isn't an integer. Eventually, someone is forced to produce a quotient of $1$, and that player loses. For example, if a player receives the number $3$, then the only prime number (s)he can possibly divide by is $3$, and this forces that player to lose.

Who must win this game, and why? Explain your reasoning in complete sentences. (Don't just show one example of how the game could go. Instead, explain why the same player must always win, no matter what strategy either player tries to use!)

Answers

Answer 1

42000 = 2^4*3*5^3*7, so has 9 prime factors

The player who plays 9th will lose. If only 2 people are playing, that is the person who takes the first turn. The second player in a 2-person game with this starting number will always win.

Related Questions

35 less than 7 times a number is 98. what is the number?

Answers

The answer is nineteen (19).Let us use this equation: 7x-35= 98x stands for the missing number.To solve for x, we will transfer the -35 to the side of 98, and it will look like this: 7x= 98+35 (NOTE: Once we transfer a number on the other side, their signs will change too. For example, the -35 became positive 35 when it was transferred to the side of the 98)Then, 7x= 133, divide both sides by 7 so that the x will remain. 133/7= 19.Therefore, x=19

Is the number of total molecules on the left side of a balanced equation always equal to the number of total molecules on the right side of the equation? explain your answer.3. is the number of total molecules on the left side of a balanced equation always equal to the number of total molecules on the right side of the equation? explain your answer?

Answers

Since no reaction creates or destroys atoms, every balanced chemical equation must have equal numbers of atoms of each element on each side of the equation. However, the number of molecules does not necessarily have to be the same.

The answer is

No, the total number of molecules can be equal or not

Answer:

No, the number of total molecules on the left side of a balanced equation is not equal to the number of total molecules on the right side of the equation. A molecule is the smallest number of atoms bonded together for a chemical reaction. The total number of atoms must be the same, but not molecules. The reactants and products will bond together in different ways leading to different numbers of reactants and products

Step-by-step explanation:

this is for pennfoster

write a polynomial (x+6)(x-2)(x-1)

Answers

Write a polynomial (x+6)(x-2)(x-1)
x3+3x2−16x+12I Hope this help

(x + 6) (x-2) (x-1)
Let's rewrite the expression step by step.
We have:
We multiply the first two parentheses:
(x ^ 2-2x + 6x-12) (x-1)
We rewrite:
(x ^ 2 + 4x-12) (x-1)
Now we multiply the remaining parentheses:
(x ^ 3 + 4x ^ 2-12x-x ^ 2-4x + 12)
We rewrite
(x ^ 3 + 3x ^ 2-16x + 12)
Answer:
The polynomial is:
(x ^ 3 + 3x ^ 2-16x + 12)

Population y grows according to the equation dy/dt=ky, where k is a constant and t is measured in years. if the population doubles every 10 years, then the value of k is

Answers

k = ln(2^(1/10)) ≈ 0.0693147

Given that the population doubles every ten years, k may be found using the population growth equation dy/dt=ky by computing ln(2)/10, or roughly 0.0693 annually.

We begin by thinking about the solution to the differential equation dy/dt = ky, where the population doubles every ten years, in order to determine the value of k. This differential equation can be solved generally as y(t) = y(0)e^kt, where y(0) is the beginning population.

Since there are ten years between population doubling, we can write: y(10) = 2y(0) = y(0)e^10k.

The result of dividing both sides by y(0) is 2 = e^10k.

We get: ln(2) = 10k by taking the natural logarithm on both sides.

After calculating k, we have k = ln(2)/10 ≈ 0.0693.

Thus, k's value is around 0.0693 per year.

Use the parabola tool to graph the quadratic function f(x)=x^2+10x+24.

Graph the parabola by first plotting its vertex and then plotting a second point on the parabol

Answers

Hello there!

Today we are just going to graph this quadratic function.

First let's find the vertex. There are two ways to find the vertex, but we will use the formula -b/2a today!

f(x)=x^2+10x+24

plug this into -b/2a where a=1 and b=10. (I found out what a and b equal by looking at the each term's coefficient.)

-10/2(1)
-10/2
-5

So the x value for our vertex is -5, so now to find the y-value, we need to plug in -5 in place of x in the equation...
f(x)=x^2+10x+24
f(-5)=(-5)^2+10(-5)+24
f(-5)=25-50+24
f(-5)=-25+24
f(-5)=-1

So the vertex is at (-5,-1)
Look at the table...
X] -5
Y] -1

Now we need to plug in number around the vertex to the equation.
X] -3 -4 -5 -6 -7
Y] -1

f(x)=x^2+10x+24
f(-4)=(-4)^2+10(-4)+24
f(-4)=16-40+24
f(-4)=-24+24
f(-4)=0

f(-3)=(-3)^2+10(-3)+24
f(-4)=9-30+24
f(-4)=-21+24
f(-4)=3

X] -3 -4 -5 -6 -7
Y] 3 0 -1

Now because the parabola is symmetrical, we don't have to plug in the other points to the equation. Watch what I do...
X] -3 -4 -5 -6 -7
Y] 3 0 -1 0 3

I really hope this helps!
Best wishes :)

Answer:

See the attachment

Step-by-step explanation:

You want a graph of the parabola defined by f(x) = x² +10x +24.

Vertex

We can add and subtract the square of half the x-coefficient to put the equation into vertex form:

f(x) = x² +10x +24 . . . . . . . . . . given

f(x) = x² +10x +25 +24 -25 . . . . add and subtract (10/2)² = 25

f(x) = (x +5)² -1 . . . . . . . . . write in vertex form

Comparing to the vertex form equation f(x) = (x -h)² +k, we see that ...

(h, k) = (-5, -1).

These are the coordinates of the vertex.

Another point

The coefficient of x² is 1, so another point can be found by adding (1, 1) to the vertex. That means a second point on the parabola is ...

(-5, -1) +(1, 1) = (-4, 0)

Graph

The attached graph shows the parabola with vertex (-5, -1) through point (-4, 0).

Additional comment

When the coefficient of x² is not 1, the process is a little different. If you start with f(x) = ax² +bx +c, you can factor 'a' from the first two terms to help you get vertex form.

f(x) = a(x² +(b/a)x) +c
f(x) = a(x² +(b/a)x +(b/(2a))²) + c - a(b/(2a))²
f(x) = a(x +b/(2a)x)² +(c -b²/(4a))

The vertex is (-b/(2a), c -b²/(4a)).

The second point can be found by adding (1, a) to the vertex coordinates.

x * 1 + x/1 = _______.

Answers

x*1+x/1
=x+x/1
=x+x
=2x

HOPE IT HELPS UH!!☺☺

X * 1 = x

x/1 = x

x + x = 2x

Your answer is 2x

Hope I helped!

Let me know if you need anything else! I love this kind of math!

~ Zoe

Suppose you have 15 days until your field trip and you need to raise $900 there are 10 students going on the field trip they will each help fundraise how much should each student have raised in 1 week?

Answers

I think the answer is $90.

Apply the distributive property to create an equivalent expression. 2(3-8y)

Answers

Answer:

An equivalent expression is [tex]6 - 16y[/tex]

Step-by-step explanation:

The distributive property consists in removing the parenthesis. For this problem, it is done by multiplying the number outside the parenthesis by each term inside the parenthesis and then performing a subtraction between the results because of the minus sign inside.

For the expression [tex]2(3-8y)[/tex], when applying the distributive property, you would get:

[tex]2\times 3 - 2\times 8y[/tex]

[tex]6 - 16y[/tex]

Thus, an equivalent expression is [tex]6 - 16y[/tex].

The distributive property to remove the parentheses in the expression 2(3 - 8y) is 6 - 16y

Using the distributive property to remove the parentheses

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

2(3 - 8y)

When the expression is expanded, we have the following

2(3 - 8y) = 2 * 3 - 2 * 8y

Evaluate the products

This gives

2(3 - 8y) = 6 - 16y

Lastly, we have

2(3 - 8y) = 6 - 16y

Hence the expression when simplified is 6 - 16y

Answers

60000=40000+100x
20000=100x
x=200

Im thinking her commission should be 1000, not 100, and if so, your equation would be

60000=40000+1000
20000=1000x
x=20

Answer:

y = $1,000/car x + $40,000

20 cars

Step-by-step explanation:

Let

x = number of cars sold

y = total earnings

The relationship between y and x can be expressed through a linear equation.

y = mx + b

where,

m is the slope

b is the y-intercept

The slope is how much she earns per car sold, that is, the commission of $1,000/car (I think you meant this number instead of $1,00).

The y-intercept is what she earns even if she does not sell any car, i.e. a salary of $40,000.

The resulting equation is

y = $1,000/car x + $40,000

If she is to make $60,000 in one year (y = $60,000), the number of cars sold is:

$60,000 = $1,000/car x + $40,000

$20,000 = $1,000/car x

x = 20 car

A ladder 10 ft long rests against a vertical wall. If the bottom of the ladder slides away from the wall at a rate of 1.3 ft/s, how fast is the angle between the ladder and the ground changing when the bottom of the ladder is 8 ft from the wall? (That is, find the angle's rate of change when the bottom of the ladder is 8 ft from the wall.)

Answers

check the picture below.

so, when r = 10 and x = 8, the vertical distance is 6, namely in pythagorean theorem lingo, b = 6.

let's keep in mind that, the ladder is not growing any longer or shrinking, and therefore is constantly always just 10, that matters, since is just a scalar value and also because the derivative of a constant is 0.

[tex]\bf cos(\theta )=\cfrac{x}{r}\implies cos(\theta )=\cfrac{1}{10}x\implies \stackrel{chain~rule}{-sin(\theta )\cfrac{d\theta }{dt}}=\cfrac{1}{10}\cdot\stackrel{chain~rule}{\cfrac{dx}{dt}\cdot 1}[/tex]

[tex]\bf -sin(\theta )\cfrac{d\theta }{dt}=-\cfrac{1}{10}\cdot \cfrac{dx}{dt}\implies \cfrac{d\theta }{dt}=-\cfrac{1}{-10sin(\theta )}\cdot \cfrac{dx}{dt} \\\\\\ \begin{cases} \frac{dx}{dt}=1.3\\ sin(\theta )=\frac{6}{10} \end{cases}\implies \cfrac{d\theta }{dt}=-\cfrac{1}{10\cdot \frac{6}{10}}\cdot 1.3\implies \cfrac{d\theta }{dt}=-\cfrac{1.3}{6}~radians[/tex]

The angle's rate of change when the bottom of the ladder is 8ft from the wall is

[tex]\dfrac{-1.3}{6} rad/s[/tex]

It is given the Length of Ladder [tex](h)[/tex] is [tex]10ft.[/tex]. and the distance between the bottom of the ladder to the wall [tex](r)[/tex] is [tex]8ft.[/tex] as shown in the below figure.

By using the Pythagoras Theorem

[tex]b=\sqrt{10^{2}-8^{2} }\\=\sqrt{36} \\=6ft.[/tex]

and

[tex]cos(\theta)= r/h\\cos(\theta)=r/10[/tex]

Differentiating both sides with respect to [tex]'t'[/tex] by using the chain rule

[tex]-sin(\theta)\dfrac{\mathrm{d}\theta }{\mathrm{d} t}=\dfrac{1}{10} \dfrac{\mathrm{d}r }{\mathrm{d} t}\\\\\dfrac{\mathrm{d}\theta }{\mathrm{d} t}=\dfrac{1}{10} \dfrac{\mathrm{d}r }{\mathrm{d} t} \dfrac{1}{-sin(\theta)} } ......(eq. 1)[/tex]

given

[tex]\dfrac{\mathrm{d}r }{\mathrm{d} t}=1.3ft./s\\sin(\theta)=\frac{6}{10}[/tex]

putting this in eq.1, we get

[tex]\dfrac{\mathrm{d} \theta}{\mathrm{d} t} = \dfrac{1.3}{10} (\dfrac{1}{-\frac{6}{10} }) \\\dfrac{\mathrm{d} \theta}{\mathrm{d} t} =\dfrac{-1.3}{6} rad/s[/tex]

So the angle's rate of change when the bottom of the ladder is 8ft from the wall is[tex]\dfrac{-1.3}{6} rad/s[/tex].

Know more about Pythagoras Theorem here:

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Larry used a pattern of colors to make a cube train he use Red Cube a blue cube a Red Cube and another Red Cube before he started the pattern again he use 15 cubes how many red cubes did Larry use

Answers

To answer this you can create the pattern up to and including the 15th term.

Red, Blue, Red, Red
Red, Blue, Red, Red
Red, Blue, Red, Red
Red, Blue, Red

Gary used 11 red cubes to make his cube train that was 15 cubes long.

find the volume of this prism

Answers

Hello!

The volume formula for a rectangular prism is length x width x height, so all we need to do is plug in the numbers.

L x W x H = V
12.5 x 4.5 x 4.2 = V
236.25 = V

Don't forget the units! We say cubed because the prism is 3-Dimensional but if it was 2-D we would say squared.

The volume is 236.25 centimeters cubed.

What is the probability that a unit chosen at random has between four and six rooms?

Answers

as probabilities says you have to put 4/6 reduce it to is lowest Dad which is equal to 2/3

the perimeter of a semi circle is 20.56 MM. What is the semicircle of radius?

Answers

Perimeter = 20.56mm

Given the perimeter, find the diameter
[tex] \pi D[/tex] = 20.56
D = 20.56 ÷[tex] \pi [/tex]
D = 6.54 mm

Perimeter of the Semicircle = half the circle arc + the straight line (Diameter)
Semicircle = 0.5(20.56) + 6.54cm
Semicircle = 16.83mm

which equation represents a proportional situation?
A. y = 9x
B. y = -2x + 23
C. y = - 3x + 4
D. y = 3x - 12

Answers

the answer is a and the it do not have y intercept

PLEASE HELP PLEASE WILL GIVE BRAINLIEST !!!!! Freya is training for a track race. She starts by sprinting 200 yards. She gradually increases her distance

Answers

Answer:

A. [tex]a_n=200+(n-1)5[/tex]

Step-by-step explanation:

Given,

The initial sprinting of Freya = 200 yards,

Also, She gradually increases her distance by 5 yards per day,

So, in second day her sprinting = 200 + 5 = 205 yards,

In third day = 205 + 5 = 210 yards,

In fourth day = 210 + 5 = 215 yards,

...................., so on,.....

Hence, the sequence that shows the given situation,

200, 205, 210, 215, .........

Which is an A.P.

That having first term, a = 200,

Common difference, d = 5,

Thus, the explicit formula for the given situation is,

[tex]a_n=a+(n-1)d[/tex]

[tex]\implies a_n=200+(n-1)5[/tex]

Option A is correct.

Answer:

Option A.

Step-by-step explanation:

The explicit formula of an AP is

[tex]a_n=a+(n-1)d[/tex] .... (1)

where,

a is the first of the AP,

d is common difference.

It is given that Freya starts by sprinting 200 yards and she gradually increases her distance, adding 5 yards a day.

200, 205, 210,..., 305

Here,

First terms = 200

Common difference = 5

Substitute a=200 and d=5 in equation (1), to find the required explicit model

[tex]a_n=200+(n-1)5[/tex]

Therefore, the correct option is A.

How do you solve 3x^3+6x^2=72x

Answers

x= -6, 0, 4
Cancel out like terms, then simplify, my dude. Hope I helped!

Oooooooooookkkkkkkkkkkkkkjjj

Can a right triangle have two angles that measure 25 and 65

Answers

Yes, because there would only be 1 right triangle and all the angles need to add up to 180 degrees and 90+25+65=180.

Find the value of each variable.

Answers

x is 19
y is 32.91
.....

A player shoots a basketball from a height of 6 feet. The equation, h = -16t 2 + 25t + 6, gives the height, h , of the basketball after t seconds. Describe the height, rounded to the nearest tenth of a foot, of the basketball after 1.5 seconds, assuming no other player touches the ball.

Answers

175t because the motion times the width gives you 175t

Answer:

The height of the ball after t =1.5 second is 7.5 feet.

Description:

The basket ball shoots from a height of 6 feet, it increase the height 1.5 feet after 1.5 seconds.

Therefore, the basketball thrown at the rate of 1 feet/per second.

Step-by-step explanation:

Given: A player shoots a basketball from a height of 6 feet. The equation,

h(t) = [tex]-16t^2 + 25t + 6[/tex]

We need to find the height of the ball when t = 1.5 and describe the height.

plug in t = 1.5 in h(t) = [tex]-16t^2 + 25t + 6[/tex]

h(1.5) = [tex]-16(1.5)^2 + 25(1.5) + 6[/tex]

Simplifying the above expression, we get

h(1.5) = -36 + 37.5 + 6

h(1.5) = -36+43.5

h(1.5) = 7.5 feet

The height of the ball after t =1.5 second is 7.5 feet

Description:

The basket ball shoots from a height of 6 feet, it increase the height 1.5 feet after 1.5 seconds.

Therefore, the basketball thrown at the rate of 1 feet/per second.

help me with this because I'm a little rusty on ths

Answers

-4 ≤ 4x + 8
----------------
Flip the equation
4x + 8 ≥ -4
----------------
Subtract 8 from each side
4x + 8 - 8 ≥ -4 - 8
4x ≥ -12
----------------
Now divide each side by 4
4x ÷ 4 ≥ -12 ÷ 4
x ≥ -3 is your answer

multiply and simplify 12 and 2 / 3 3 + 1 / 4

Answers

(12 2/3)*(3 1/4) = 41 1/6 . . . . . according to my calculator

_____
You are usually taught to multiply the improper fractions and simplify the result.
.. (38/3)*(13/4) = (38/4)*(13/3)
.. = (19/2)*(13/3)
.. = (19*13)/6
.. = 247/6
.. = 41 1/6

You can also multiply the parts.
.. (12 2/3)*(3 1/4) = 12*3 + 12*(1/4) +(2/3)*3 +(2/3)*(1/4)
.. = 36 +3 +2 +1/6
.. = 41 1/6

∆FGH , the measure of <H=90, FH=48,GF=73,and, HG=55 What is the value of the sine of <F to the nearest hundredth

Answers

sin(F) = HG/GF = 55/73 ≈ 0.75

Find dy/dx by implicit differentiation. 8 cos x sin y = 6

Answers

8sin(x)cos(y) = 6
Take derivative with respect to x. Since y is a function of x, take the derivative for y as well but it is multiplied by dy/dx

chain rule
8cos(x)cos(y) - 8sin(x)sin(y)(dy/dx) = 0

solve for dy/dx

8cos(x)cos(y) = 8sin(x)sin(y)(dy/dx)

[8cos(x)cos(y)]/[8sin(x)sin(y)] = dy/dx
simplify
cot(x)cot(y) = dy/dx

Steven, a tailor, got an order to make a blazer. The customer specifically asked him to save 5/6 of a foot of the given cloth to make a pocket square. However, Steven accidentally saved 5/12 of a foot. What is the difference between the requested cloth and the saved cloth? A. 0.1466' B. 0.4166' C. 0.4265' D. 04066'

Answers

The answer to your question is B. The 6 is forever repeated fyi...

The sum of 5 consecutive numbers is 135

Answers

27+27+27+27+27 is 135

Answer:

Step-by-step explanation:

The answer is 25+26+27+28+29=135

Is the correct answer I basically divided 135 by 5 and got 27 then I just worked around that number to get the answer.

Which is the best approximation for the solution of the system of equations

Answers

x = 0.882
y = 0.647
seems to be a good approximation of the solution.

Answer:

Solution of the system of the equations is (0.882, 0.647)

Step-by-step explanation:

As shown in the figure equations are [tex]y=-\frac{2}{5}x+1[/tex] and y = 3x - 2

Solution of the system of equations will be the common point or point of intersection of these lines.

To get the point of intersection we will solve these equations.

We will equate these equations to get the value of x.

[tex]-\frac{2}{5}x+1=3x-2[/tex]

[tex]-2x+5=15x-10[/tex]

15x + 2x = 10 + 5

17x = 15

[tex]x=\frac{15}{17}[/tex]

x = 0.882

By putting x = 0.882 in y = 3x - 2

y = 3(0.882) - 2

y = 0.647

So the solution of systems of equations is (0.882, 0.647)

Find the general solution of the given second-order differential equation. y'' − y' − 30y = 0 webassign

Answers

y''-y'-30y=0;
1) the characteristic equation is:
a²-a-30=0, where a²⇒y'', a⇒y' and 1⇒y.
[tex] \left[\begin{array}{ccc}a=6 \\a=-5\end{array}[/tex]
2) y=C₁*e⁶ˣ+C₂*e⁻⁵ˣ

The general solution of the given second-order differential equation

y'' - y' - 30y = 0 is,

⇒ y = C₁ e⁶ˣ + C₂ e⁻⁵ˣ

What is mean by Function?

A relation between a set of inputs having one output each is called a function. and an expression, rule, or law that defines a relationship between one variable (the independent variable) and another variable (the dependent variable).

Given that;

The second-order differential equation is,

⇒ y'' − y' − 30y = 0

Now, We can simplify as;

⇒ y'' − y' − 30y = 0

This gives the general form as;

⇒ m² - m - 30y= 0

⇒ m² - 6m + 5m - 30 = 0

⇒ m (m - 6) + 5 (m - 6) = 0

⇒ (m + 5) (m - 6) = 0

⇒ m = - 5 or m = 6

Hence, The general solution of the given second-order differential equation y'' - y' - 30y = 0 is,

⇒ y = C₁ e⁶ˣ + C₂ e⁻⁵ˣ

Learn more about the function visit:

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Which equations represent inverse variation? Check all that apply.
y = 2x
pv = 13
z = (2/x)
4 = (y/x)
h = (9g/5)

Answers

For this case, what you should know is that the equations that represent an inverse variation are those that could not form a straight line, for example.

We have then:
Equation 1:
pv = 13
Rewriting:
p = 13 / v
P and v are represent an inverse variation.
Equation 2:
z = (2 / x)
z and x are represent an inverse variation.
Answer:
equations represented inverse variation are:
pv = 13
z = (2 / x)

Answer:

B and C

Step-by-step explanation:

I don't understand this at all

Answers

Try this option:
if to re-write and simplify the given expression, then answers:
the coeffic. for a²= -2;
the coeffic. for ab= -2+6=4;
the coeffic. for b=6-8= -2.

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