NO LINKS!! Please help me with this problem. Part 8ff​

NO LINKS!! Please Help Me With This Problem. Part 8ff

Answers

Answer 1

Answer:

[tex]\dfrac{1}{36n^2+6n}[/tex]

Step-by-step explanation:

Given factorial expression:

[tex]\dfrac{(6n-1)!}{(6n+1)!}[/tex]

[tex]\boxed{\begin{minipage}{6cm}\underline{Factorial Rule}\\\\$n!=\:n\cdot \left(n-1\right) \cdot \left(n-2\right) \cdot ... \cdot 3 \cdot 2\cdot 1$\\ \end{minipage}}[/tex]

Apply the factorial rule to the numerator and denominator of the given rational factorial expression:

[tex](6n-1)!=\left(6n-1\right)\cdot \left(6n-2\right)\cdot \left(6n-3\right)\cdot... \cdot 3 \cdot 2\cdot 1[/tex]

[tex]\left(6n+1\right)!=\left(6n+1\right)\cdot \:6n \cdot (6n-1) \cdot...\cdot 3 \cdot 2\cdot 1[/tex]

Therefore:

[tex]\begin{aligned}\implies \dfrac{(6n-1)!}{(6n+1)!}&=\dfrac{\left(6n-1\right)\cdot \left(6n-2\right)\cdot \left(6n-3\right)\cdot... \cdot 3 \cdot 2\cdot 1}{\left(6n+1\right)\cdot \:6n \cdot (6n-1) \cdot...\cdot 3 \cdot 2\cdot 1}\\\\&=\dfrac{1}{(6n+1) \cdot 6n}\\\\&=\dfrac{1}{6n(6n+1)}\\\\&=\dfrac{1}{36n^2+6n}\end{aligned}[/tex]

Answer 2

Answer:

[tex]\cfrac{1}{6n(6n+1)}[/tex]

--------------------------------

We know that:

n! = 1·2·3·4·...·n

Therefore:

(6n + 1)! = (6n - 1)!·6n·(6n + 1)

Therefore:

[tex]\cfrac{(6n-1)!}{(6n+1)!} =\cfrac{(6n-1)!}{(6n-1)!(6n)(6n+1)} =\cfrac{1}{6n(6n+1)}[/tex]

Related Questions

Calculate two-thirds of three-quaters of one half

Answers

Answer:

0.25

Step-by-step explanation:

Step 1: Convert all fractions to decimals

Two-thirds = 0.6666

Three-quarters = 0.75

One-half = 0.5

Step 2: Multiply all fractions

0.6666 * 0.75 * 0.5 = 0.25

Step 3: The answer is 0.25

Answer:

[tex]\dfrac{1}{4}[/tex]

Step-by-step explanation:

Three-quarters of one half:

[tex]\implies \dfrac{3}{4} \times \dfrac{1}{2}=\dfrac{3 \times 1}{4 \times 2}=\dfrac{3}{8}[/tex]

Two-thirds of "Three-quarters of one half":

[tex]\implies \dfrac{2}{3} \times \dfrac{3}{8}=\dfrac{2 \times 3}{3 \times 8}=\dfrac{6}{24}=\dfrac{6 \div 6}{24 \div 6}=\dfrac{1}{4}[/tex]

What is the equation, in stanard form, of a parabola that models the values in the table x-3 0 2 f(x) -8 -2 -28

Answers

The equation of the parabola is f(x)= 7x²+5x-2 when values of the x are -3,0,2 and the f(x) are -8,-2,-28.

Given that,

The values of the x are -3,0,2 and the f(x) are -8,-2,-28

We have to find the equation of the parabola with the x and f(x) values.

We know that,

The equation of the parabola of the form f(x) = ax²+bx+c

So,

-3a-3b+c=-8 ----->equation(1)

c=-2

2a+2b+c=-28 ------>equation(2)

Substituting c=-2 in equation(1) and equation(2)

-3a-3b-1+8=0

-3a-3b+7=0 ------>equation(3)

2a+2b-2+28=0

2a+2b+26=0 ------>equation(4)

Subtracting 2×equation(3) and 3×equation(4)

                               b=5

Substituting in equation(1)

a=7

Therefore, The equation of the parabola is f(x)= 7x²+5x-2 when values of the x are -3,0,2 and the f(x) are -8,-2,-28.

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Parallel and Perpendicular lines Block

Answers

By observing the figure we can make the pairs as

∠1 ≅ ∠3 ---------(Opposite angles)
∠2 ≅ ∠4 ---------(Opposite angles)
∠5 ≅ ∠7 ---------(Opposite angles)
∠6 ≅ ∠8 ---------(Opposite angles)
∠4 ≅ ∠6 ---------(Alternate angles)
∠3 ≅ ∠5 ---------(Alternate angles)

What are opposite angles and Alternate angles?

Opposite angles are angles that are located opposite each other, with respect to a straight line or a transversal. They are also called vertically opposite angles.

Alternate angles are angles that are located on opposite sides of the transversal and on the same side of the straight line or the transversal. They are also called corresponding angles.

In the given figure,

By observing the opposite angles and alternate angles we can fill the blocks,

∠1 ≅ ∠3 ---------(Opposite angles)

∠2 ≅ ∠4 ---------(Opposite angles)
∠5 ≅ ∠7 ---------(Opposite angles)
∠6 ≅ ∠8 ---------(Opposite angles)


∠4 ≅ ∠6 ---------(Alternate angles)

∠3 ≅ ∠5 ---------(Alternate angles)

Hence, we have found the opposite angles and alternate angles from the figure.

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Using traditional methods, it takes 100 hours to receive a basic driving license. A new license training method using Computer Aided Instruction (CAI) has been proposed. A researcher used the technique with 260 students and observed that they had a mean of 99 hours. Assume the standard deviation is known to be 6. A level of significance of 0.01 will be used to determine if the technique performs differently than the traditional method. Find the value of the test statistic. Round your answer to 2 decimal places.

Answers

The value of the Test statistic =z=-1.67

We have the value of sigma = 6, n = 260

In a hypothesis test, a test statistic—a random variable—is computed from sample data. To decide whether to reject the null hypothesis, you can utilize test statistics. Your results are compared to what would be anticipated under the null hypothesis by the test statistic. The p-value is computed using the test statistic.

Now using the formula of Test static

Test statistic will be:  [tex]\mathrm{z}=\frac{\overline{\mathrm{x}}-\mu}{\frac{\sigma}{\sqrt{\mathrm{n}}}}\\[/tex]

Where, n = total number of sample = 100

[tex]\bar x[/tex] = mean of the given data

[tex]\mu[/tex] = total number of hours taken to receive a basic driving licence.

So, putting the value given in the question,

We will get the value of test static as:

[tex]=\frac{99-100}{\frac{6}{\sqrt{260}}}=-1.67[/tex]

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Find x to the nearest tenth.

Answers

After using trigonometric function the value of to the nearest tenth digit is 11.11

What is Trigonometric function ?

Trigonometric functions are mathematical functions that are used to describe relationships between certain angles and lengths in a triangle. They are based on the ratios of the sides of a right triangle and are used in a variety of applications including surveying, navigation, engineering, and physics. Trigonometric functions are commonly referred to as sine, cosine, tangent, cotangent, secant, and cosecant. Each of these functions can be used to calculate the angle of a triangle, the length of the sides, or the area of the triangle.

Function of Sine

The ratio of the opposing side's length to the hypotenuse is known as an angle's sine function.
Sin a =Opp/Hypot = CB/CA

Cos Function
The cosine of an angle is the ratio of the neighbouring side's length to the hypotenuse's length.
Adjacent/Hypotenuse = AB/CA is the cosine of.

Functional Tan
The ratio of the lengths of the adjacent and opposing sides is known as the tangent function. It should be noted that the ratio of sine and cosine to the tan may also be used to express the tan.
Tan a = Opposite/Adjacent = CB/BA
In terms of sine and cos, tan is also equivalent to:
Tan = cos a/sin a
Sin 46° = 8/x
x = 8/Sin 46°
x= 8/0.72
x=11.11

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What are the coefficients in the following expression x^2+2x-5xy-y+3y^4

Answers

Answer:

1,2,-5,-1,3

Step-by-step explanation:

the coefficients are the numbers before the variables (x and y)

if there is nothing infront of the variable then the coefficient is 1

if there is only a negative sign infront of the variable then the coefficient is -1

convert the following equation into standard form.
Y= -2x+6

Answers

Answer:

x = 3+y

Step-by-step explanation:

is your ans .................

Isosceles trapezoid ABCD is shown.
A
B
D
C
Which three statements are correct?
ZBAC ZCAD
ZACD ZACB
ZADC ZBCD
AD
AB
AC
BC
CD.
BD
ہے

Answers

From the given Isosceles trapezoid ABCD, the three correct statements are:

∠ADC ≅ ∠BCD

AD ≅ BC

AC ≅ BD

What is Isosceles trapezoid?

An isosceles trapezoid is a trapezoid with equal base angles and hence equal left and right side lengths.

Particularly in isosceles trapezoids, there are unique correlations. "Equal legs" is what the term "isosceles" denotes. Non-parallel sides on isosceles trapezoids have the same lengths. The "legs" are another name for these equal sides.

A triangle with two equal sides is said to be isosceles. Also equal are the two angles that face the two equal sides. A triangle with two congruent sides is referred to as being isosceles, in other words.

Thus, three correct statements are -

∠ADC ≅ ∠BCDAD ≅ BCAC ≅ BD

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What is the equation of the circle shown on the graph

Answers

The equation of the circle shown on the graph is (x + 2)² + (y - 1)² = 4.

What is an equation of a circle?

A circle can be characterized by its centre's location and its radius's length.

Let the center of the considered circle be at the (h, k) coordinate.
Let the radius of the circle be 'r' units.
Then, the equation of that circle would be:
(x - h)² + (y - k)² = r²
The center of the circle is at (-2, 1) and the radius of the circle is 2 units. Then the equation of the circle will be
(x - (-2))² + (y - 1)² = 2²

        (x + 2)² + (y - 1)² = 2²

        (x + 2)² + (y - 1)² = 4
Hence, the equation of the circle shown on the graph is (x + 2)² + (y - 1)² = 4.

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find the value of x in this problem

Answers

Answer: x=22

Step-by-step explanation:

Based on the given figure, we can tell that 2x and 44 are vertical angles. Vertical angles are congruent to each other, so we can set them equal to each other.

2x=44           [divide both sides by 2]

x=22

Therefore, x=22.

Express in a single log term

Answers

The simplified form of the given logarithmic expression as required in the task content bis Choice B; 2log8.

What is the simplified form of the expression as a single log term?

It follows from the task content that the simplified form of the logarithmic expression is to be determined as a single log term.

On this note, since the given expression is;

2log2 + 2log4

It follows from factorisation that we have;

2 ( log 2 + log 4 )

Therefore, the expression in the parentheses can be simplified as a single log term using the sum of logarithms law as follows;

= 2 ( log ( 2 × 4 ) )

= 2 log 8.

Ultimately, the answer choice which represents the simplified form of the expression as a single log term is; Choice B; 2 log 8.

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Annette's Floral Shop has a charge account with Southern Flowers, a wholesaler where she
buys her floral supplies. This month, Annette has purchased 16 dozen roses at $30.96 per
dozen, 40 dozen chrysanthemums at $17.80 per dozen, and 20 dozen tulips at $21.10 per
dozen. In her state, Annette must pay a 6% sales tax on all orders. At the end of the month,
what is the balance on Annette's charge account?

Answers

Total balance in Annette's charge account is $1727.12 with 6% of tax on all order.

What is Percentage ?

The Latin phrase "per centum," which meaning "by the hundred," was the source of the English word "percentage." Fractions having a denominator of 100 are called percentages. In other words, it is a relationship where the value of the entire is always assumed to be 100.

Finding the percentage of a whole in terms of 100 is what percentage calculation refers to. A percentage can be found in one of two ways:

use the unitary approach.by adjusting the fraction's denominator to 100.

It should be noted that when the denominator is not a factor of 100, the second technique of percentage calculation is not applied. In these situations, we employ the unitary technique.

1 dozen of Rose costs = $30.96

1 dozen of chrysanthemums costs = $17.80

1 dozen of of tulips costs = $ 21.10

Total Spending = 16 dozens of roses + 40 dozen of chrysanthemums + 20 dozens of tulips

                         = 16 * 30.96 + 40 * 17.80 + 20 *21.10

                         = $1629.36

There is a 6% sales tax on all order = 6% of 1629.36 =  $ 97.76

Total balance in Annette's charge account is $ 97.76 + 1629.36 = $1727.12.

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Total balance in Annette's charge account is $1727.12 with 6% of tax on all order.

What is Percentage ?
The Latin phrase "per centum," which meaning "by the hundred," was the source of the English word "percentage." Fractions having a denominator of 100 are called percentages. In other words, it is a relationship where the value of the entire is always assumed to be 100.

Finding the percentage of a whole in terms of 100 is what percentage calculation refers to. A percentage can be found in one of two ways:

Use the unitary approach.
by adjusting the fraction's denominator to 100.
It should be noted that when the denominator is not a factor of 100, the second technique of percentage calculation is not applied. In these situations, we employ the unitary technique.
1 dozen of Rose costs = $30.96
1 dozen of chrysanthemums costs = $17.80
1 dozen of of tulips costs = $ 21.10
Total Spending = 16 dozens of roses + 40 dozen of chrysanthemums + 20 dozens of tulips
                        = 16 * 30.96 + 40 * 17.80 + 20 *21.10
                        = $1629.36
There is a 6% sales tax on all order = 6% of 1629.36 =  $ 97.76
Total balance in Annette's charge account is $ 97.76 + 1629.36 = $1727.12.

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The sum of an infinite geometric series with first term a and common ratio r < 1 is given by The sum of a given a/1-r infinite geometric series is 300, and the common
ratio is 0.1. What is the second term of this series?

Answers

The second term of the series will be 27.

What is a Geometric progression?

Geometric Progression (GP) is a type of sequence where each succeeding term is produced by multiplying each preceding term by a fixed number, which is called a common ratio. This progression is also known as a geometric sequence of numbers that follow a pattern.

Sum to infinity = a/1-r

where s = 300

r = 0.1

a = 300 (1 - 0.1)

a = 300 (0.9)

a = 270

The second term of the progression will be = ar

Second term = 270 x 0.1

Second term = 27

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What is the answer for identifying the slope and Y intercept for this graph ?

Answers

Slope is -2 and intercept is 3.
The y intercept is where the equation meets the y axis

The table and grab below shows the distance travel between two different objects which statement is correct about the speed of the two objects

Answers

The statement which is correct about the speed of the two objects is both travel at steadily increasing speed.

What is meant by direct relation?

A direct relation exists when the two variables rise or fall together. In a straight link, as X rises, Y rises as well. A straight relationship will always have a positive slope on a graph.

What is speed?

Speed is the pace at which an object's position changes in any direction. The distance travelled in relation to the time it took to travel that distance is how speed is defined. Since speed simply has a direction and no magnitude, it is a scalar quantity.

The graph shows the direct relation that means with increasing time and distance speed will increase. Moreover, the table also shows direct and increasing relation about the speed of two objects so the statement both travel at steadily increasing speed is correct.

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How could you organize the weighting and grades information differently so that Oscar's and Reagan's final grades are given by WG?

Answers

Answer i just need points

Step-by-step explanation: maybe 90

Sketch the curve with the given vector equation by finding the following points. (Select Update Graph to see your response plotted on the screen. Select the Submit button to grade your response.) r(t) = (t, 9 – t, 2t) r(-9) (x, y, z) r(0) (x, y, z) r(9) (x, y, z)

Answers

For sketching the graph we need to know the coordinate of the vector which are r(1) = ⟨−1, 2⟩, r′(1) = ⟨1, 2⟩

Given:-

x = t − 2,

y = t² + 1 ⇒ t = x + 2

⇒ y = (x + 2)2 + 1

⇒ y = x² + 4x + 5

For find r′(t) we have to ,

r′(t) = ⟨1, 2t⟩

To sketch the position vector r(t) and the tangent vector r′(t) for t = 1.

r(1) = ⟨−1, 2⟩, r′(1) = ⟨1, 2⟩

See the above figure. The red vector is r(1) and the blue one is r′(1)

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Given the ellipse with equation substitute the x-values from the table into the equation to obtain y-values, rounded to the nearest integer.

Answers

The value of y when X value is -1 would be = -5

What is substitution equation method?

The substitution equation method is the method of solving equation whereby a value is being simplified and substituted into the second equation to obtain the next unknown value.

From the given equation:

(x-2)²/16 - (y-4)²/9 = 1

Take X = -1 and substitute X for -1 into the given equation,

(-1-2)²/16 - (y-4)²/9 = 1

9/16 - (y-4)²/9 = 1

(y-4)²/9 = 9/16- 1

(y-4)²/9 = -7/16

Cross multiply

(y -4)² = 9(-7)/16

y²+16 = -63/16

16(y²+16) = -63

16y²+256 = -63

16y² = -319

y² = -319/16

y² = -20

y= -√20

y = -4.5

y = -5

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100 POINTS
Find the focus. x^2 = 6y (please explain, I want to understand the solution)

Answers

The coordinate of the parabola's focus in the following equation, x2=6y, is (0,3/2)

What are Directrix and Focus?

What are a parabola's focus and directrix? Typically speaking, parabolas are the graphs of quadratic functions. They can alternatively be thought of as the collection of all points whose separation from the focus is the same as their separation from a certain line (the directrix).

What are the focus's coordinates?

The focus's coordinates f(0,a)

The equation that we have given is

x^2 = 6y

We are aware that the parabola's general equation is,

x^2 = 4ay

The axis of symmetry is a line that splits the parabola in half and runs through the focus.

We must ascertain the worth of

6y = 4ay

6 = 4a

a = 6/4 = 3/2

The focus' coordinate (0,3/2) is as a result.

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Find the vertex of the graph of f(x) = |0.25x − 0.75|. vertex ?

Answers

The vertex of the graph (3, 0)

What is Vertex of graph?

A vertex or node is the basic building block of a graph in discrete mathematics, and more precisely in graph theory. An undirected graph is made up of a set of vertices and a set of edges, whereas a directed graph is made up of a set of vertices and a set of arcs.

According to question

when f(x) = 0

that's the minimum point, the vertex, or the intersection of two lines,  g(x) = 0.25x − 0.75 and h(x) =  0.75 - 0.25x

Therefore

find the intersection of Two line which are

⇒ y = x/4 - 3/4

⇒ 4y = x - 3 → First line

⇒ y = 3/4 - x/4  

⇒ 4y = 3 - x → Second line

So x - 3 = 3 - x

2x = 6 ,

x = 3

And y = 0

hence the vertex of f(x) = (3, 0)

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Which functions have a vertex with a x-value of 0? Select three options.
Of(x) = |x|
f(x) = x + 3
f(x) = 1x + 31
f(x) = 1x1-6
Of(x) = x + 31-6

Answers

The functions that have a vertex at x = 0 are (a) f(x) = |x| and (d) f(x) = |x| - 6

How to determine the function from the vertex

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

The absolute functions in the options

As a general rule;

A absolute functions can be represented as

f(x) = a|x - h| + k

Where

Vertex = (h, k)

A vertex that has x value of 0 means

(h, k) = (0, k)

So, we have

f(x) = a|x - 0| + k

Evaluate

f(x) = a|x| + k

Looking at the options, we have

f(x) = |x| and f(x) = |x| - 6

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What is the greatest common factor of 18, 24, and 40?

2

3

4

8

Answers

Answer:

  (a)  2

Step-by-step explanation:

You want the greatest common factor of 18, 24, 40.

Factors

The prime factorization of these numbers is ...

18 = 2 · 3²24 = 2³ · 340 = 2³ · 5

The only common factor is 2.

2 is the greatest common factor of 18, 24, and 40.

t=10-x identify to dependent and independent variable

Answers

Answer: t is the dependent variable  and x the independent variable

Step-by-step explanation: t is the dependent variable it depends upon the values of x that we put in.

x is the independent variable as we can use whatever values we want for x it is not affected by t

PLEASE HELP IM GOING TO FAIL


4. What is the graph of the system? (1 point)
y≤x +4
2x+y≤-4

Answers

Answer:

A

Step-by-step explanation:

The answer is option A

Hope this helps :)

given a binary-class classification problem in which the class labels are binary, the dimension of feature is d, and each attribute can take k different values. please provide the numbers of parameters to be estimated with and without the simplifying assumption. please explain your answer. briefly justify why the simplifying assumption is necessary.

Answers

For binary class classification, assumptions are necessary in a way:

Given that

From Boyer Naive classifier,

We evaluate (ai | vj) is given below

(ai | vj) = n(e) + m(p) / n + m

Here,

m = equivalent sample size

n(e) = number of training examples

for which v = vj and a = ai

n(i) = number of training example for which v = vj

P = a prior estimate for P(ai | vj)

Here, we calculate that

P(SUV | yes), P(Red | yes), P(Domestic | yes)

P(SUV | no), P(Red | no), P(Domestic | no)

We evaluating this value like

yes:

RED :                             SUV:                            DOMESTIC:

n = 5                              n = 5                             n = 5

n(c) = 3                          n(c) = 1                           n(c) = 2

P = 0.5                          P = 0.5                           P = 0.5

m = 3                              m = 3                            m = 3

Therefore, with the above calculation we can justify that the simplifying assumptions are necessary in a binary class classification.

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PLEASE ANSWERE ASAP

If BC = 16.7 ft, what is AY?

Answers

Answer:

Since YA=AX, XB=BZ, ZC=CY, it follows that A, B and C are the midpoints of the sides TX, XZ, ZY. AB, BC, AC are parallel to the sides and equal to their half. In fact, AB = YC=ZC. Therefore BC=AY.)

Good luck;)

Step-by-step explanation:

Question 1
Two months ago, a car dealership sold 30 red cars and 120 cars that were not red. Last
month the car dealership sold a total of 200 cars. Tarik predicts that the dealership sold
45 red cars last month. Do you think this is a reasonable prediction? Explain.

Answers

Answer:

Yes this is a resendable prediction

Step-by-step explanation:

Given the drawing as shown below and that p parallel q, which of the following cannot be supported?​

Answers

Answer: It's B

Step-by-step explanation:

It shows that Angle B is congruent to angle g but it actually isnt

Answer:

B.  ∠b ≅ ∠g

Step-by-step explanation:

Supplementary angles

Two angles whose measures sum to 180°.

Linear Pair

Two adjacent angles that sum to 180° (two angles which when combined together form a straight line).

Same-side Interior Angles Theorem

When two parallel lines are intersected by a transversal, the angles that are interior to the parallel lines and on the same side of the transversal line are supplementary (sum to 180°).

Vertical Angles Theorem

When two straight lines intersect, the opposite vertical angles are congruent.

------------------------------------------------------------------------------------------

Statement A

∠f and ∠h are vertical angles.  Therefore, according to the Vertical Angle Theorem, ∠f ≅ ∠h.

Statement C

As ∠d and ∠h are the interior angles on the same side of the transversal [tex]\ell[/tex], according to the Same-side Interior Angles Theorem, ∠d and ∠h are supplementary.

Statement D

∠a and ∠b are a linear pair, therefore ∠a and ∠b are supplementary.

Therefore, Statement B cannot be supported by the evidence shown.

You are making snack bags to sell at the fair. You want each bag to contain the same combination of carrot and celery sticks. You have 75 carrot sticks and 40 celery sticks and want to use them all.

What is the greatest number of snack bags you can make

Answers

The greatest number of snack bags is 10.

Highest Common Factor:

The largest of all the common factors between two or more numbers is known as the Highest Common Factor (HCF). It is also known as the Greatest Common Element (GCF).

HCF is a fundamental method that allows you to equally distribute items throughout a group or set.

We can use the idea of HCF to determine the same when you are organizing a party and want to make sure that nothing is wasted or you require a proper calculation.

Here we have

Number of carrots = 75

Number of celery sticks = 40

Here we are making snack bags such that each bag contains the same combination of carrot and celery sticks.

To find the greatest number of snack bags we need to find HCF of 70 and 40 as given below

Write given numbers as the product of prime numbers

=> 70 = 2 × 5 × 7

=> 40 = 2 × 2 × 2 × 5  

List out the common factors

=> 2 × 5  

=> HCF (70, 40) = 10

Therefore,

The greatest number of snack bags is 10.

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Let f:R→S be a surjective homomorphism of rings with identity.
(a) If R is a PID, prove that every ideal in S is principal.
(b) Show by example that S need not be an integral domain.

Answers

Every ideal of S is principal when f:R⇒S be a surjective homomorphism of rings with identity.

In a homomorphism, corresponding elements of two systems behave very similarly in combination with other corresponding elements. For example, let G and H be groups. The elements of G are denoted g, g′,…, and they are subject to some operation ⊕.

In algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type (such as two groups, two rings, or two vector spaces). The word homomorphism comes from the Ancient

Let f:R⇒S be a surjective homomorphism of rings with identity.

We have to find if R is a PID, prove that every ideal in S is principal.

We know that,

Let I be the ideal of S

Since f is sufficient homomorphism.

So, f⁻¹(I) is an ideal of R.

Since R is PID so ∈ r ∈ R such that

f⁻¹(I) = <r>

I = <f(r)>

Therefore,

Every ideal of S is principal when f:R⇒S be a surjective homomorphism of rings with identity.

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