Prove that an integer consisting of 3n ones is divisible by 3n (e.g. 111 is divisible by 3, 111,111,111 is divisible by 9, etc.) Prove it by induction. Notice, that while the number with a sum of digits divisible by 3 is itself a multiple of 3 and the number with a sum of digits divisible by 9 is itself a multiple of 9, it is NOT true that the number with a sum of digits divisible by 27 is necessarily a multiple of 27.

Answers

Answer 1

For proof of 3 divisibility, abc is a divisible by 3 if the sum of abc (a + b + c) is a multiple of 3.

Integers divisible by 3

The proof for divisibility of 3 implies that an integer is divisible by 3 if the sum of the digits is a multiple of 3.

Proof for the divisibility

111 = 1 + 1 + 1 = 3  (the sum is multiple of 3 = 3 x 1)  (111/3 = 37)

222 = 2 + 2 + 2 = 6 (the sum is multiple of 3 = 3 x 2)  (222/3 = 74)

213 = 2 + 1 + 3 = 6 ( (the sum is multiple of 3 = 3 x 2)  (213/3 = 71)

27 = 2 + 7 = 9  (the sum is multiple of 3 = 3 x 3)  (27/3 = 9)

Thus, abc is a divisible by 3 if the sum of abc (a + b + c) is a multiple of 3.

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Related Questions

If x and y are complementary angles, then y is a right angle.

right?

Answers

Answer: No

Step-by-Step Explanation:

Complementary angles are a set of angles that sum up to 90 degrees. Neither y nor x can be 90 since if it is, then the other angle would have to be 0 degrees which is not possible in a triangle.

What is the quotient if -3/8 and -1/3

Answers

[tex]\left(- \dfrac 38 \right) \div \left(-\dfrac 13 \right)\\\\\\=-\dfrac 38 \times (-3)\\\\\\=\dfrac 98[/tex]

What is the measure of major B 84° O 276° O252° O 84° O 168 arc AB? A​

Answers

Answer:

A - 84°

Step-by-step explanation:

The measure of an arc is equal to the measure of the central angle that intersects it.

The central angle of this circle measures 84° and it intercepts arc AB, meaning arc AB measures 84°

given that у=6 cm and θ=15 °, work out x rounded to 1 decimal place.

(the diagram is not drawn to scale.)

Answers

Using a trigonometric realtion, we can see that x = 23.18cm

How to find the value of x?

Here we have a right triangle, we can see that y = 6cm, and the angle theta is 15°.

Notice that y is the opposite cathetus to the known angle, and we want to find the value of x, which is the hypotenuse.

Then we can use the trigonometric relation:

sin(a) = (opposite cathetus)/hypotenuse.

Reaplacing what we know, we will get:

sin(15°) = 6cm/x

Solving that for x:

x = 6cm/sin(15°9

x = 23.18cm

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A bottle contains 1 2/5 litres of orange squash. To make one drink, 1/200 of a litre of squah is needed. How many drinks can be made from the bottle of squash?

Answers

Answer:

280 drinks

Step-by-step explanation:

Capacity of bottle [tex]=1\frac{2}{5}=\frac{7}{5}\: l[/tex]

Squash needed to make one drink [tex]=\frac{1}{200}\:l[/tex]

Total number of drinks [tex]=\frac{7}{5}\div\frac{1}{200}[/tex]

[tex]=\frac{7}{5}*200[/tex]

[tex]=7*40[/tex]

[tex]=280[/tex]

Answer:

280 drinks. We need to divide 1 2/5 by 1/200 in order to get this answer.

                                                     

Please help me with this probability question; I’d really appreciate it!

The attendance at a village fair depends on the weather.
The probability of a high attendance at the fair is 0.9.
This is reduced to 0.4 if it is raining.
The probability of it raining is 0.2.
What is the probability of its raining and there being a high attendance?
0.6
0.08
0.36
0.18

Answers

Answer:

B. 0.08

Step-by-step explanation:

In order to solve this problem, we must account for all possibilities. The sum of all posssiblities must sum up to 1.

1 - High Attendance/No Rain

0.9*0.8 = 0.72

2- High Attendance/ Rain

0.4*0.2 = 0.08

3 - Low Attendance/No Rain

0.1*0.8 = 0.08

4 - High Attendance/No Rain

0.6*0.2 = 0.12

Now we answer the question of rain and high attendance.

0.08/1 = 0.08

Find the general solution of the given differential equation. Cos(x) dy dx (sin(x))y = 1

Answers

The general solution to the differential equation is y = sinx + c cosx.

What is differential equation?

It is defined as the equation which contains functions, derivative of the functions to make an expression.

We have a differential equation:

[tex]\rm cosx\left(\dfrac{dy}{dx}\right)+\left(sinx\right)y\:=\:1[/tex]

Multiply by dx:

(cosx) dy + (sinx) y dx = dx

(cosx) dy = (1 - (sinx) y) dx

After simplification and integrating, we will get:

[tex]\rm y = \dfrac{tan x}{sec x}+ \dfrac{c}{secx}[/tex]

y = sinx + c cosx

Thus, the general solution to the differential equation is y = sinx + c cosx.

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Please help me with this!

Answers

Answer:

sin(4)

Step-by-step explanation:

sin(-184) is the equivalent to sin(4) (degree), positive and acute angle.


The perimeter of an equilateral triangle is 96 cm. Find the length of the side of the triangle.

Answers

Answer:

Answer : Length(l) = 32 cm

Step-by-step explanation:

Given,

          Perimeter(P)=96 cm

          Length(l)=?

We know that,

                        P      = 3l

                        96    = 3l

                        96÷3 = l

                         32    = l

Therefore, length(l) = 32 cm

Find the volume of the following figure.

6 cm3
7 cm3
8 cm3
5 cm3

Answers

Answer:

7 cm³

Step-by-step explanation:

Assume each cube has an edge 1 cm in length.

Each cube has volume 1 cm³.

There are 7 cubes.

volume = 7 cm³

6x1 + 9 x 1/10 + 8 x 1/100 + 3 x 1/1000=?

Answers

Answer:

6983/1000 OR 6.983

Step-by-step explanation:

6*1=6

9*1/10=9/10

8*1/100=8/100

3*1/1000=3/1000

6+9/10+8/100+3/1000=6983/1000

OR

6.983

What is the solution of 5sin2θ=3 for 0≤θ≤2π? Please help.

Answers

Answer:

C) [tex]\theta\approx\{0.3218,1.249,3.463,4.3908\}[/tex]

Step-by-step explanation:

[tex]5\sin2\theta=3;\: 0\leq\theta\leq 2\pi\\\\\sin2\theta=\frac{3}{5}\\ \\2\theta\approx\{0.6436,2.498,6.926,8.7816\}\\\\\theta\approx\{0.3218,1.249,3.463,4.3908\}[/tex]

For every 36 minutes of sprinting, Scarlett earned 12 dollars for charity. To find the unit rate of this, I would first need to structure the equation. In this case, I would use division, so my equation would be 36 ÷ 12 = 3. The final answer would be, for every 3 minutes, Scarlett earns $1. is this right? i will give 15 points:)

Answers

Answer:

$1 every 3 minutes

Step-by-step explanation:

Cause every 36 minutes Scarlett earns $12 and if you divide 36 by 12 it will give 3 minutes which means she earns $1 per 3 minutes. You're right.

What is the value of x log3x equals 4

Answers

[tex]~~~~\log_3 (x) = 4\\\\\implies x= 3^4 \\\\\implies x= 81[/tex]

How do I graph this linear function
x < -1

Answers

Hope this helps! Let me know if you need anything else.

This is how the linear function looks.

A large bin of Peanut M&M's contains red, yellow, blue, and brown M&M's. Lexi
scoops out a bowl full of M&M's and counts how many of each type she has: 12 red,
7 yellow, 14 blue, and 9 brown.
Luke scoops out a bowl of 36 Peanut M&M's. About how many should he expect to
be blue?

Answers

Answer:

About 12 blue m&m’s

Step-by-step explanation:

30 Use the method of completing the square to determine the vertex of f(x) = x² - 14x - 15.
State the coordinates of the vertex.

Answers

Answer:

x² - 14x - 15 =  (x − 7)² − 64Then the coordinates of the vertex are (7 , -64)

Step-by-step explanation:

x² - 14x - 15 = x² - 14x + (49 - 49) - 15

                  = (x² - 14x + 49) - 49 - 15

                  =  (x² - 14x + 7²) - 49 - 15

                  =  (x − 7)² − (49 + 15)

                  =  (x − 7)² − 64

x² - 14x - 15 =  (x − 7)² − 64

Then the coordinates of the vertex are (7 , -64)

The mean mass of a squad of 19 rugby players is 87 kg
A player of mass 102 kg joins the squad.
Work out the mean mass of the squad now, in kilograms (kg).

Answers

19 players x 87kg = 1653 total kg

1653 kg + 102kg = 1755 kg

1755kg / 20 players = 87.75kg mean mass

Answer: 87.75 kg

5600 to the nearest 1000​

Answers

Answer:

6000.

Step-by-step explanation:

Hundreds digit is 6 so we round up the thousand digit (5) to get 6000.

Figure FGHJ is shown below.
Figure F G H J has 4 sides. Sides F J and G H are congruent and parallel, and sides G F and H J are congruent and parallel. Angles G and J are 130 degrees.

Which names accurately describe figure FGHJ? Select two options.
parallelogram
quadrilateral
rectangle
rhombus
trapezoid

Answers

The correct answer is a parallelogram which has both pairs of opposite sides parallel to one another.

A random number generator is used to create a list of 200 single-digit numbers. of those 200 numbers, 105 are even and ninety-five are odd. seven was generated fifteen times. what is the experimental probability of an odd number other than seven being generated? 40.0% 45.0% 47.5% 52.5%

Answers

Using it's concept, it is found that the probability of an odd number other than seven being generated is of 40.0%.

What is a probability?

A probability is given by the number of desired outcomes divided by the number of total outcomes.

For an experimental probability, these number of outcomes are taken from previous trials.

In this problem, 200 numbers have been generated, and 80 are odd numbers that are not seven, hence the probability is given by:

p = 80/200 = 0.4 = 40%.

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Answer:

40

Step-by-step explanation:

PLS I NEED 100% TO GET A GOOD MATH GRADE THIS YEAR
The following box plot shows the number of years during which 24 schools have participated in an interschool swimming meet:

At least how many schools have participated for 1 year or less? (5 points)



6 schools


8 schools


12 schools


14 schools

Answers

Answer:

6 schools

Step-by-step explanation:

I did the same assignment

Explain the difference between 2(7x-4) and (7x-4)^2

Answers

2(7x-4) is the equation being multiplied by two and (7x-4)^2 is the equation being multiplied by itself twice

Answer:

2(7x-4) = 7x(2)-4(2) = 14x-8

but

(7x-4)^2 = (7x-4) (7x-4)

5² + 12² =
64a3³
Work out the value of a.
Your final line should say, a =

Answers

Answer:

a = 13/4

Step-by-step explanation:

√(5²+12²) = ∛(64a³)

√(25+144)=√(64a)

√169 = √(64a)

13 = 4a

a = 13/4

What is the y intercept of this quadratic? (Picture)

Answers

Answer:

We know that the y intercept is -3 because that is where it intersects! The point for the y intercept is (0, -3)

Step-by-step explanation:

This is the coordinates of the point wher the graph cuts the y axis It is (0,-3)

Rotate the point (7, 8) around the origin 90 degrees counter clockwise. State the image of the point .

Answers

By applying the equation for the rotation around the origin we conclude that the image of the point (7, 8) after rotating 90° counterclockwise is (-8, 7).

How to rotate a point around the origin

In this question we must apply a rigid transformation on a point to find its image, rigid transformations are transformations applied on geometric loci such that Euclidean distances are observed in every point of the loci. A rotation about the origin is a kind of rigid transformation and is defined by the following expression:

(x', y') = (x · cos θ - y · sin θ, x · sin θ + y · cos θ)

Where:

(x, y) - Original point(x', y') - Resulting pointθ - Angle of rotation, in degrees.

Please notice that positive values of θ represents a counterclockwise rotation. If we know that (x, y) = (7, 8) and θ = 90°, then the resulting point is:

(x', y') = (7 · cos 90° - 8 · sin 90°,  7 · sin 90° + 8 · cos 90°)

(x', y') = (-8, 7)

By applying the equation for the rotation around the origin we conclude that the image of the point (7, 8) after rotating 90° counterclockwise is (-8, 7).

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I need help! 50 points!

Answers

The equation of this sinusoidal function is either

f(x) = a sin(bx) + c

or

f(x) = a cos(bx) + c

Either way, the plot of f9x) has amplitude a, period 2π/b, and midline y = c.

If the period is π/2, then

2π/b = π/2   ⇒   b = 4

If the maximum value is 10 and the minimum value is -4, then

-4 ≤ a sin(4x) + c ≤ 10

-4 - c ≤ a sin(4x) ≤ 10 - c

-(4 + c)/a ≤ sin(4x) ≤ (10 - c)/a

Recall that sin(x) is bounded between -1 and 1. So we must have

-(4 + c)/a = -1   ⇒   a = c + 4

(10 - c)/a = 1   ⇒   a = -c + 10

Combining these equations and eliminating either variable gives

a + a = (c + 4) + (-c + 10)   ⇒   2a = 14   ⇒   a = 7

a - a = (c + 4) - (-c + 10)   ⇒   0 = 2c - 6   ⇒   c = 3

Finally, we have either

f(x) = a sin(bx) + c   ⇒   f(0) = c = 3

or

f(x) = a cos(bx) + c   ⇒   f(0) = a + c = 3

but the cosine case is impossible since a = 7.  

So, the given function has equation

f(x) = 7 sin(4x) + 3

Multiply the following
(a) 2.7x4
(b) 0.79x32.4
(c)1.07 x 0.02
(d) 2.3x4.35
(e) 2.5x100
(f) 0.0008x100
g) 0.25x 10000
h)3x0.55
i)0.08x5
j)351.06x4
k)0.3x0.03
l) 21.7x0.1
m) 200.02 x 2.2

Kindly look into this and try to answer all of them this contains 30 points

Answers

Answers:

a) 10.8
b) 25.596
c) 0.0214
d) 10.005
e) 250
f) 0.08
g) 2500
h) 1.65
i) 0.4
j) 1404.24
k) 0.009
l) 2.17
m) 440.044

Note: I haven’t provided the steps since this is basic multiplication of decimal numbers. However, if you do need help please comment below this answer.

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PLS HELP
Which shows the following expression after the negative exponents have been eliminated?

Answers

Answer:

Option 2

Step-by-step explanation:

Given :

[tex]\frac{xy^{-6} }{x^{-4}y^{2} }[/tex]

=============================================================

Applied Rule :

[tex]\frac{x^{a} }{x^{b} } = x^{a-b}[/tex]

============================================================

Solving :

⇒ To remove negative exponents, bring to the opposite side (e.g. if in the denominator, bring to the numerator)

⇒ [tex]\frac{x(x^{4}) }{y^{2}(y^{6}) }[/tex]

help i will give 20 thax

Answers

Ty's brother will get 3/4th part of the pizza.

What is Division ?

The division is a method of distributing a group of things into equal parts. It is one of the four basic operations of arithmetic, which gives a fair result of sharing.

Here, Total number of Pizza = 1

         Number of parts required = 4

Now, each member will get the part of pizza = 1/4

Total number of brothers = 3

Ty's brother will get pizza = 3 X 1/4

                                           = 3/4

Thus, Ty's brother will get 3/4th part of the pizza.

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Answer:

3/4

Step-by-step explanation:

because there are 4 people so ty get 1/4 and his brothers get the rest.

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