Mathematics Middle School

Prove the divisibility of the following numbers:

51^7-51^6 by 25

Answers

Answer 1

To prove the divisibility lets expand each number first to get the complete picture of the problem

51^7 =   8.974106779 . 10^11

51^6 =  1.75962878 . 10^10

(51^7 - 51^6) =  8.974106779 . 10^11 -  1.75962878 . 10^10

                    = 8.798143901. 10^11

[tex]\frac{8.798143901. 10^11}{25}[/tex]

= 3.51925756. 10^10

That is the answer expressed in scientific notation

= 35192575600

Which means the divisibility is proven

Answer 2

Step-by-step explanation:

[tex]51^7-51^6=51^{6+1}-51^6\\\\\text{use}\ a^n\cdot a^m=a^{n+m}\\\\=51^6\cdot51^1-51^6\cdot1\\\\\text{use the distributive property}\ a(b-c)=ab-ac\\\\=51^6\cdot(51-1)=51^6\cdot50=51^6\cdot(2\cdot25)=25\cdot(51^6\cdot2)[/tex]

[tex]\text{One of the factors of the product is the number 25.}\\\\\bold{CONCLUSION:}\\\\\text{The whole product is divisible by 25.}[/tex]

[tex]\text{Therefore}\ 51^7-51^6\ \text{is divisible by 25.}[/tex]

Related Questions

The image on a movie poster was shrunk to make the DVD cover art for the movie, so that the cover art is a scale image of the poster. The poster is 36 inches wide, and the DVD cover art is 6 inches wide. If the diagonal of the poster is 4 feet, what is the diagonal of the DVD cover art?

Answers

Please see the attached image for a visual representation of our scale factor. We can set up this proportion by taking the DVD cover and poster values and placing them in fractions. Cross multiply and divide to solve for x. 
The first thing we need to do is find the proportionality constant [tex]k[/tex]. Since we now that the width of a movie poster was shrunk from 36 inches to 6 inches to make the DVD cover, [tex]k= \frac{36}{6} =6[/tex]; therefore, the width of poster was shrunk, in other words divided, by a factor of 6. 
Now that we have our proportionality constant lest apply it to the diagonal of the poster to find the diagonal of the DVD cover. But notice that the diagonal of the poster is in feet not inches, so we need to convert 4 feet to inches first. Remember that 1 foot = 12 inches, so lets use that proportion to set up our conversion fraction:
[tex](4feet)( \frac{12inches}{1 foot} )= 48inches[/tex]
The only thing left now is shrunk, divide, the diagonal by our proportionality constant [tex]k[/tex]
[tex] \frac{48}{k} = \frac{48}{6} =8[/tex]

We can conclude that If the diagonal of the poster is 4 feet, the diagonal of the DVD cover will be 8 inches.

A meteor crater is 3900 feet in diameter. Approximate the distance around the crater. Use 3.14 for π.

Answers

You’re looking for the *circumference* of the crater. The definition of π is the ratio of the circumference of the circle to its diameter, or mathematically:

π = c/d

Which means that c = dπ. Our diameter is 3900 ft, which mean that our circumference is 3900π ft, or, using 3.14 as π, 3900(3.14) = 12,246 ft.

Quadrilateral ACEG≅Quadrilateral NMRP Which statement is true?
A:
EG≅MN
B:AC≅MR
C: AG≅NP
D:CE≅NP
pls help

Answers

Final answer:

The true statement given that quadrilateral ACEG is congruent to quadrilateral NMRP is D: CE≡NP, which matches the corresponding sides based on the order of the vertices in the name of each quadrilateral following CPCTC.

Explanation:

If quadrilateral ACEG is congruent to quadrilateral NMRP, then corresponding parts of congruent polygons are congruent as well. This principle is often abbreviated as CPCTC (Corresponding Parts of Congruent Triangles are Congruent). Applying this rule, we need to match each vertex of quadrilateral ACEG to the corresponding vertex of quadrilateral NMRP.

To determine which statement is true, we need to compare the corresponding parts:

EG≡MN: This would be true if G corresponded to N and E to M, which would make EG and MN opposite sides.AC≡MR: This would imply that A corresponds to M and C to R, which are not opposite each other based on the naming of the quadrilaterals. This is not correct.AG≡NP: For this to be true, A would have to correspond to N and G to P. But again, A and G are not opposite vertices so this can't be true.CE≡NP: If C corresponds to N and E corresponds to P, then CE and NP would be equivalent and therefore congruent. Since there are four sides, the connecting letter should signify that C corresponds to N, and E to P, making this the correct correspondence.

Hence, the correct statement based on the information given and the application of CPCTC is D: CE≡NP.

The first person who answers correctly gets thanks and braniy!

1. The figures shown are similar. What is the scale factor? [Image one]


2. The figures shown are similar. What is the scale factor? [Image 2]


Answers

That arrow helps a lot.
One
Scale factor = 21/14 = (3*7)/(2*7) = 3/2 the sevens cancel out.

Two
Scale factor = 24/18 = (6 *4) / (6 * 3) The 6s cancel out 
Scale factor = 4/3

find the measures of the numbered angles in each figure

Answers

1 is 90o
2 is 90 - 27 = 63
3  63 angles 2 and 3 are in an isosceles triangle. They are opposite equal sides so the angles are equal.

What is the area, in square feet, of the shaded region?
What is the area, in square feet, of the nonshaded region?

Answers

Draw a line going from the intersection of the 2 foot line going up and the 2 foot line going across. Draw this line so it is parallel to the 16 foot line at the bottom.

The orange figure is now cut into 2 pieces. The bottom piece is a rectangle and the top piece is a trapezoid.

Trapezoid
========
The area of a trapezoid = (a + b)*h/2; a and b are the parallel lines. h = 2 + 8 = 10 feet
a = 1
b = 8 - 2 -2 = 4 feet. You are going to have to draw the diagram carefully to get this answer.
a is on the right side of the trapezoid. b is on the left side.
h = 8 + 2 = 10 feet

Area = (1 +4)*10/2 =25 square feet.

Rectangle
=======
L= 8 feet
w = 2 feet
A = L * w
Area= 2 *8 = 16 square feet.

Total Area Orange
==============
Area = 16 + 25 = 41 square feet

Area of the Black Area 
=================
Area = L*W - 41 Square feet
Area =16 *8 - 41 = 87 square feet

Elmo buckets made 54 out of 80 free throw .What percent did he miss

Answers

Answer:

67.50%

Step-by-step explanation:

We can prove that the answer is correct by taking 67.50 percent of 80 to get 54:

(80 x 67.50)/100 = 54

Note that our calculator rounds the answers up to two decimals if necessary. Once again, the answer is as follows:

67.50%

A rope that is 2 feet and 6 inches long is cut into 3 strips in a ratio of 2: 3: 5 How long is the longest piece         How much longer is the longest piece than the shortest one?

Answers

The longest piece is 15 inches i think 

What is the equation of the line in slope-intercept form?

[ fill in the spaces ]
y=___x+___

Answers

y=5/2x+5 because the slope is 5/2 and the y-intercept is 5

What is the median value of the data set shown on the line plot?

Answers

If there is an odd number of observations, the median is the middle one, if we order the observations in increasing order. If the number is even, then we take the mean of the two middle ones. Here, we see that we have in total 22 observations, so we take the mean of the 11th and 12th observation. We have that 9 observations are below 6. Then, the 10th until the 13th observation is 6. Hence, we have that the median is [tex] \frac{6+6}{2} =6[/tex].

The answer is 6. I'm just adding this to help anyone else struggling with this quiz.

Please help offering All my pts and brainliest answer


(06.02 LC)Which of the following algebraic expressions can be used to express the statement "one less than x squared?"

1 − x2

x2 − 1

1 − 2x

2x − 1

Answers

x2-1 I think that's the answer because there's 1 less, 1 take away the x and squared!! hard to explain but that's how I thought about it

Algebraic expression can be used to express the statement one less than x squared is option (B) [tex]x^{2} -1[/tex]

What is Algebraic expression?

In mathematics, an algebraic expression is an expression built up from integer constants, variables, and the algebraic operations.

Given,

one less than x squared is

[tex]x^{2} -1[/tex]

Hence, the algebraic expression can be used to express the statement one less than x squared is option (B) [tex]x^{2} -1[/tex]

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1.2 decreased by 40%

Answers

1.2 x 0.4 = 0.48
1.2 - 0.48 = 0.72
Solution = 0.72

I hope this helps
0.72 is the answer.

Each side of the square below is 20 centimeters. The area of the circle inside the square is about 314 square centimeters.

a.43/200
b.43/157
c.157/200
d.200/157

Answers

Maybe divide each number until you get 20?

Answer:

It would be 157/200

Step-by-step explanation:

The gent in the comments


Please help!!!

What sum of money will yield a simple interest of Rs.1155 in 3 years 6 months at 11% per annum .

Answers

Answer:

Rs.833.94

Step-by-step explanation:

To solve for the sum of money or principal, we use the formula:

[tex]P=\dfrac{A}{1+rt}[/tex]

The current variables we have are:

A = Rs.1,155

r = 11% or 0.11

t = 3.5

Now let's plug the variables into the formula.

[tex]P=\dfrac{A}{1+rt}[/tex]

[tex]P=\dfrac{1,155}{1+(0.11)(3.5)}[/tex]

[tex]P=\dfrac{1,155}{1+0.385}[/tex]

[tex]P=\dfrac{1,155}{1.385}[/tex]

[tex]P=Rs.833.94[/tex]

The distance around a circle is 56.55 meters long. What is the diameter of the circle?

Answers

the answer is 113.1 I hope it helps you and I hope this is correct.

Answer: Diameter of circle = 18m

Explanation:

Given that distance around a circle is = 56.55 meter

The distance around a circle on the other hand is called the circumference .

That means Circumference = 56.55 m

We know the formula of circle's circumference= 2πr

where r is the radius and 2r is diameter of the circle

So,

2πr = 56.55

divide by π both the side

2r = [tex] \frac{56.55*7}{22} [/tex]

Value of π = [tex] \frac{22}{7} [/tex]

2r = 17.993 ≈ 18 m

Diameter of circle = 18m

In the figure, point B is the midpoint of AC. Use the figure to answer the questions.

(a) Jeremy says that ABD is congruent to CBD by the SAS congruence postulate. Do you agree or disagree? Explain.


(b) Suppose it is also known that AD is congruent to CD. Which postulate or theorem can be used to prove that ABD is congruent to CBD? Justify your answer.

Answers

Answer:

A is true

B = SSS OR RHS

Step-by-step explanation:

bcoz both can be applied here

Jeremy's statement that triangle ABD is congruent to triangle CBD by the SAS congruence postulate is incorrect.

If we are also given that AD is congruent to CD, then we can use the SSS congruence postulate to prove that triangle ABD is congruent to triangle CBD.

Answer to part (a)

I disagree with Jeremy's statement that triangle ABD is congruent to triangle CBD by the SAS congruence postulate.

The SAS congruence postulate states that two triangles are congruent if they have two pairs of corresponding sides congruent and the included angle congruent.

In this case, we have two pairs of corresponding sides congruent (AB = BC and AD = CD, since B is the midpoint of AC). However, we are not given that the included angle (angle ABD) is congruent to angle CBD.

Therefore, we cannot use the SAS congruence postulate to prove that triangle ABD is congruent to triangle CBD.

Answer to part (b)

If we are also given that AD is congruent to CD, then we can use the SSS congruence postulate to prove that triangle ABD is congruent to triangle CBD.

The SSS congruence postulate states that two triangles are congruent if they have three pairs of corresponding sides congruent.

In this case, we have three pairs of corresponding sides congruent (AB = BC, AD = CD, and BD = BD). Therefore, we can use the SSS congruence postulate to prove that triangle ABD is congruent to triangle CBD.

For more such information on: SAS congruence

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which translation maps the graph of the function f(x)=x^2 onto the function g(x)=x^2-6x+6

Answers

write x²-6x+6 into vertex form by completing a perfect square:
x²-6x+9+6-9=(x-3)²-3
compared with f(x)=x², g(x)=(x-3)²-3 is shifted three units to the right and three units down. 

Answer:

Translation from (0,0) to (3,-3), 3 units down, and 3 units to the right.

Step-by-step explanation:

The Geometric Transformation called Translation of this parabola f(x)=x² is obtained, firstly by shifting it down 3 units simply adding -3 units as the independent term.

The Translation of the Parabola is counted by its vertex. y=x² (0,0) to y=x²-6x+6 (3,-3).

Algebraically:

So, f(x)=x² -3

Looking at g(x)=x²-6x+6, we still need to make it down after the curve intercept y-axis. b<0

Then, let's shift it three units to the right so its vertex hit the point 6, as y-coordinate, we have to add -3. If we want to add to the right sum negative value, then square it:

y= (x-3)²-3

y=x²-6x+9-3

y=x²-6x+6

HELP EXPERTS AND GENIUSES PLS HELP

What is the best approximation for the circumference of a circle with a radius of 45 cm?

Use 3.14 to approximate pi.

1:70.65 cm
2:141.3 cm
3:282.6 cm
4:6358.5 cm

Answers

45 * 2 = 90
90 x 3.14 = 282.6 cm

 answer is: 3:282.6 cm

Answer:

282.6

Step-by-step explanation:

you have to find the volume of this figure

Answers

area of lower triangular prism:
1/2 x L x H X W
1/2 x 8 x 4.5 x 30 = 540

area of top triangular pyramid
1/3 x B x H
B = 1/2 bxh = 1/2 x 8 x 4.5 = 18

1/3 x 18 x 10 = 60

total volume = 540 + 60 = 600 cubic cm

An economy package of cups has 200 green cups. If the green cups are 10% of the total package, how many cups are in the package?

Answers

Final answer:

Set up a proportion where 10% of the total number of cups equals 200 green cups, then calculate to find that there are 2000 cups in the entire package.

Explanation:

To find the total number of cups in the package when 200 green cups represent 10% of the total, we can set up a simple proportion. Here, 10% corresponds to the number of green cups we know about. We need to find 100% to discover the total number of cups in the package.

The proportion we'll set up is:

10% / 200 cups = 100% / x cups

To solve for x (the total number of cups), we will divide 200 cups by 10% to find the value of 1% and then multiply by 100%:

1% of total = 200 cups / 10

x (total cups) = (200 cups / 10) × 100

Perform the calculation:

x = 20 cups × 100

x = 2000 cups

So, there are 2000 cups in the total package.

A navigator plots the course a plane is currently traveling. The plane is 300 miles from its destination. If it continues on its current course it will travel 325 miles and end up 125 miles due south of its destination. To the nearest degree, how many degrees is the plane off course?

Answers

We have to use the rule of cosx° to solve this problem. Attached is a diagram of the navigator's course for the plane. It is similar to the shape of a triangle. We know the plane is 300 miles from its destination, so that will be one of the sides. On the current course, it is 325 miles from its destination, so that will be another one of the sides. The last side is 125 because that is the distance between the destination and the anticipated arrival. Cosx° is what we are looking for. 

To find how many degrees off course the plane is, we must use the rules of Cosx°, which is shown in the attached image. 

The plane is approximately 23° off course. 
Final answer:

The plane in question is approximately 23 degrees off course. This calculation is derived from the tangent function in trigonometry based on the distances given.

Explanation:

From the question, we understand that the plane is creating a right triangle due to its off-course direction. Using the Pythagorean theorem, the length of the hypotenuse, or the distance that the plane will travel, is 325 miles. The difference between the direct route (300 miles) and the off-course southern distance (125 miles) forms the two other sides of the triangle.

However, we're looking for the angle, not the sides. This is where trigonometry comes in. Specifically, the tangent function will be valuable here as it involves the opposite and adjacent sides of a right triangle (the off-course southern distance and the direct route distance). If we let the 'off-course' angle be θ, then we have:

Tan(θ) = (distance south / direct distance) = (125 miles / 300 miles)

In order to find the angle θ from this equation, we'll need to use the inverse tangent function.

θ = atan(125/300) = 22.61986495 degrees.

Therefore, to the nearest degree, the plane is approximately 23 degrees off course.

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1/ 4 of a pizza is split equally between 3 sisters. How much of a whole pizza will each sister get?

Answers

i think it is 1/12 of the pizza
you divide 1/4 by 3 which is 1/12

A triangular prism and two nets are shown:






Which is the correct net of the prism and what is the surface area of the prism?

Answers

the correct net of the prism is Net A
surface area of the prism is 
Surface area = Area of two triangular faces + Area of bottom rectangle + Area of two side rectangles 

View attachment

Hope this helps

The correct net of the prism is net A and the surface area of the prism is 510 square inches

How to determine the surface area

To calculate the area of the triangular prism, we make use of the net A

Start by calculating the area of the rectangle.

The dimension of the rectangle is (5 + 12 + 13) by 15

So, the area of the rectangle is:

[tex]Area = (5 + 12 + 13) * 15[/tex]

[tex]Area = 450[/tex]

Next, calculate the area of the two congruent triangles of base 5 and height 12

[tex]Area = 2 * 0.5 * (5 * 12)[/tex]

[tex]Area = 60[/tex]

Sum both areas

[tex]Total = 450 + 60[/tex]

[tex]Total = 510[/tex]

Hence, the surface area of the prism is 510 square inches

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Help with this question please!!

Answers

The first one is an answer because it simplifies to:
[tex]3(6x) = 18x[/tex]

The third one is answer because it simplifies to:
[tex]2x(9) = 18x[/tex]

Those are all your answers: Option 1 and Option 2.

A trapezoid has an area of 24 square meters. The length of the bases of the trapezoid are 5 meters and 7 meters. What is the height of the trapezoid?

Answers

As with any formula, fill in the values you know and solve for the one you don't know.

A = (b1 +b2)/2*h
24 m^2 = (5 m +7 m)/2 * h
24 m^2 = (6 m) * h
(24 m^2)/(6 m) = h
4 m = h

The height of the trapezoid is 4 meters.

Explain one relationship between capacity and fluid ounces

Answers

1 US gallon = 128 fluid ounces

Michaela caught 3 fish, which weigh a total 19 1/2 pounds. One fish weighs 7 5/8 pounds and another weighs 5 3/4 pounds. How much does the third fish weigh?

Answers

Find a common denominator for 2,4,8. Which is 8. Change the fractions to new ones.

1) 19 4/8= 156/8 Then add the second. 2) 7 5/8= 61/8 one and the third
3) 5 6/8= 46/8 one. Which is

15/8. Then, subtract your answer and the first one. You get 141. Simplify. 17 5/8

6 and 1/8. Tell me if you would like to see my equation.

Over several months, a meteorologist recorded a total snowfall of 8.6 centimeters. During this period, the average monthly snowfall was 4.3 centimeters. For how many months did the meteorologist collect measurements of the snowfalls?

Answers

He recorded the snowfall for 2 months

Which of the following represents the most accurate estimation of 375+147

Answers

375+147=522
rounded to the nearest hundred is 500

Gabe filled 5 glasses of the same size with ice tea from a full 64-ounce pitcher. There were 3 ounces left in the pitcher. Which equation and solution give the number of ounces, s, in each glass?

Answers

Answer:

The number of ounces in each glass is [tex]12.2\ ounces[/tex]

Step-by-step explanation:

Let

s-------> the number of ounces in each glass

we know that

To get the number of ounces in each glass, subtract the amount of tea left in the pitcher from the total amount of tea and divide the result by the number of glasses.

so

The equation that represent the situation is

[tex]s=(64-3)/5[/tex]

[tex]s=(61)/5=12.2\ ounces[/tex]

Answer:

Step-by-step explanation:

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