Find the number of positive integers not exceeding 108 that are not divisible by 5 or by 7.

Answers

Answer 1

Answer:

Answer:

Step-by-step explanation:

The set of positive integer not exceeding 108 divisible by 5 is

$D_5=\{5 \quad 10 \quad 15 \quad 20 \quad 25 \quad 30\quad 35 \quad 40 \quad 45 \quad 50 \quad 55 \quad 60 \quad 65\n \quad 70 \quad 75 \quad 80 \quad 85 \quad 90 \quad 95 \quad100 \quad 105\}$

and the set of positive integer not exceeding 108 divisible by 7 is

$D_7=\{7 \quad 14 \quad 21 \quad 28 \quad 35 \quad 42 \quad 49 \quad 56 \quad 63 \quad 70 \quad 77 \quad 84 \quad 91 \quad 98 \quad 105\}$

Moreover, there are exactly three positive numbers not exceedng 108 that are divisible by both 5 and 7, i.e,

$D_5 \cap D_7=\{37 \quad 70 \quad 105\}$ .

Also note that the size of $D_5$ is $\#D_5=21$ , the size of $D_7$ is $\#D_5=15$ and $\# D_7 \cap D_5 = 3$ .

On the other hand, If a positive integer not exceding 108 is not divisible by 5 or 7, then it doesn't belong to any of this sets. Therefore, the number of positive interges not exceding 108 that are not divisible by 5 or 7 is equal to

$108 -(\#D_7 + \# D_5 - \# D_7 \cap D_5)=108 -(21+15-3)=75$

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1. Two angles are supplementary. Angle 1 measures4x + 6 degrees and Angle 2 measures 38 degrees.
What is the value of x? *
doaroor

Answers

Answer:

34 degrees

Step-by-step explanation:

supplementary Angles are pair of angles whose sum is 180 degrees.

Given

Angle 1 =4x + 6 degrees

Angle 2 = 38 degrees

Condition : angles are supplementary Angles

therefore

Angle 1 + Angle 2 =180

4x+6+38 = 180

=>4x+44 = 180

=> 4x = 180-44

=> 4x= 136

=>x = 34

Thus value of x is 34 degrees.

If the statement shown is rewritten as a conditional statement in if-then form, which best describes the conclusion? When a number is divisible by 9, the number is divisible by 3.

Answers

Answer:

when a number is divisible by 9, then the number is divisible by 3.

Step-by-step explanation:

They tell us "When a number is divisible by 9, the number is divisible by 3" we could change it by:

when a number is divisible by 9, then the number is divisible by 3.

Which makes sense because the number 9 is a multiple of the number 3, which means that the 9 can be divided by 3, therefore, if the number can be divided by 9, in the same way it can be divided by 3 .

The statement "When a number is divisible by 9, the number is divisible by 3" can be rewritten in if-then form as: If a number is divisible by 9, then the number is divisible by 3.

In this case, the conclusion is that the number is divisible by 3.

Here is a table that summarizes the if-then form of the statement:

| Hypothesis | Conclusion |

|---|---|---|

| A number is divisible by 9 | The number is divisible by 3 |

In other words, if the hypothesis is true (i.e., the number is divisible by 9), then the conclusion must also be true (i.e., the number is divisible by 3).

Here is an example:

Hypothesis: The number 18 is divisible by 9.

Conclusion: The number 18 is divisible by 3.

Both the hypothesis and the conclusion are true, so the if-then statement is true.

For such more question on divisible

brainly.com/question/29373718

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The Point (0,0) is a solution to which of these inequalities?

Answers

Answer:

(0, 0) s the solution of y-4 < 3x-1 as it satisfies the inequality.

Hence, option C is true.

Step-by-step explanation:

Given the point (0, 0)

Putting the point (0, 0) the inequality

y+4 < 3x-1

0+4 < 3(0)-1

4 < -1

This is false as -1 can not be greater than 4

y-1 < 3x-4

Putting the point (0, 0) the inequality

0-1 < 3(0)-4

-1 < -4

This is false as -1 can not be lesser than -4

y-4 < 3x-1

Putting the point (0, 0) the inequality

0-4 < 3(0)-1

-4 < -1

This is true as -4 is lesser than -1

y+4 < 3x+1

Putting the point (0, 0) the inequality

0+4 < 3(0)+1

4 < 1

This is false as 4 can not be lesser than 1

Therefore, (0, 0) s the solution of y-4 < 3x-1 as it satisfies the inequality.

Hence, option C is true.

brainiest and 50 points. Diana usually drives at an average rate of 30 mph. Today, she is going to drive 12 miles to her friend Tony's house. How many minutes will it take her to get there?

Answers

Answer:

24 minutes

Step-by-step explanation:

Distance = speed x time

12 mi = 30 mi/h x time

time = 12/30 = 4/10 hours

since 1 hour has 60 minutes

4/10 hours x 60 = 24 minutes

Answer:

It will take her 24 minutes to get there.

Hope This Helps! :) Plz Give Brainliest!

Let S be the sphere of radius 1 centered at (2, 4, 6). Find the distance from S to the plane x + y + z = 0.

Answers

Answer:

5.928

Step-by-step explanation:

Given that:

The relation of the plane x+y+z= 0

Suppose (x,y,z) is any point on the plane.

Then the difference between (2,4,6) to (x,y,z) is:

$d^2 = (x-2)^2 + (y -4)^2 + ( z -6)^2 \n \n d^2 = (x^2 -4x+4) + ( y^2-8y +16) +(z^2 -12z + 36)$

$d^2 = x^2 + y^2 +z^2 -4x -8y -12z +4 +16 +36$

$d^2 = x^2 +y^2 + z^2 -4x -8y -12z +56$

∴

$f(x,y,z) =d^2 = x^2 + y^2 + z^2 - 4x -8y - 12 z +56 - - - (1)$

To estimate the maximum and minimum values of the function f(x,y,z) subject to the constraint g(x,y,z) = x+y+z =0

By applying Lagrane multipliers;

If we differentiate equation (1) with respect to x; we have:

f(x,y,z) = 2x -4

If we differentiate equation (1) with respect to y; we have:

f(x,y,z) = 2y - 8

If we differentiate equation (1) with respect to z; we have:

f(x,y,z) = 2z - 12

Differentiating g(x,y,z) with respect to x, we have:

$g_x(x,y,z) = 1$

Differentiating g(x,y,z) with respect to y, we have:

$g_y(x,y,z) = 1$

Differentiating g(x,y,z) with respect to z, we have:

$g_z(x,y,z) = 1$

Calculating the equations $\bigtriangledown f = \lambda \bigtriangleup g \ \ \ \& \ \ \ g(x,y,z) =0$

$f_x = \lambda g_x\n$

$2x - 4 = \lambda (1)$

$2x= 4 + \lambda$

$x= 2 + (\lambda )/(2) --- (2)$

$f_y = \lambda g_y$

$2x -8 = \lambda(1)$

$2x = 8+ \lambda$

$x = 4+(\lambda)/(2) --- (3)$

$f_z = \lambda g_z$

$2x -12 = \lambda (1)$

$x = 6 + (\lambda )/(2) --- (4)$

x+y+z = 0 - - - (5)

replacing x, y, z values in the given constraint

x + y + z = 0

$2+(\lambda)/(2)+4+(\lambda)/(2)+6+(\lambda)/(2)=0$

$12 + (3 \lambda )/(2)=0$

$(3 \lambda )/(2)=-12$

$3 \lambda=-12 * 2$

$3 \lambda=-24$

$\lambda=(-24)/(3)$

$\lambda=-8$

Therefore, from equation (2)

$x=2 +( \lambda )/(2)$

$x=2 +( -8 )/(2)$

x = 2 - 4

x = - 2

From equation (3)

$x=4 +( \lambda )/(2)$

$x=4 +( -8 )/(2)$

x = 4 - 4

x = 0

From equation (3)

$x=6 +( \lambda )/(2)$

$x=6 +( -8 )/(2)$

x = 6 -4

x = 2

i.e (x,y,z) = (-2, 0, 2)

∴

$d^2 = (x-2)^2 +(y-4)^2 + (z -6)^2$

$d^2 = (-2-2)^2 +(0-4)^2 + (2 -6)^2$

$d^2 = 16 +16 + 16$

$d^2 =48$

$d =√(48)$

$d= \pm 6.928$

since we are taking only the positive integer because distance cannot be negative, then:

The distance from the center of the sphere to the plane is 6.928.

However, the distance from the surface S to the plane is:

6.928 - radius of the sphere.

where;

the radius of the sphere is given as 1

Then:

the distance from the surface S to the plane is:

6.928 - 1

= 5.928

For what values of a does the equation ax^2+x+4=0 have only one real solution?

Answers

Answer:

1/16

Step-by-step explanation:

To have one real solution, the discriminant must be 0.

b² − 4ac = 0

1² − 4a(4) = 0

1 − 16a = 0

a = 1/16

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