1. Prove the following properties of divisibility for integers a, b, c, d, m, and n:

(a) if a|b and a|c, then a|mb + nc

b) if a|b and c|d, then ac|bd
2. Use the division algorithm to show that an integer of the form 3k − 1 for k ∈ Z is never a perfect square.

3. Show that if a, b ∈ N are both odd, then a
4 + b
4 − 2 is divisible by 16.

4. If n > 4 is composite, then n divides (n − 1)!.

5. Each integer n > 11 can be written as the sum of two composite integers.

Analyze the case and opinion in the case of Riser v. QEP Energy, 777 F.3d 1191 (10th Cir. 2015) located in your textbook on pp. 451–454. SCREENSHITS OF CASE STUDY BELOW

Write a case study review of Judge Kelly’s opinion that answers the questions below. Support your review with analysis and evidence from the unit reading and outside sources.

What are the legal issues presented in this case? Did the plaintiff establish a valid prima facie case of pay discrimination? Was there a basis for equal work? On what factors did the appeals court base its decision?

Why is the other-than-sex factor that is presented by the employer insufficient to avoid a trial?

What should the employer have done differently?

Your paper should be a minimum of two pages, not including the title and reference pages. You are required to use a minimum of three sources, one of which may be your textbook as reference material for your case study. Adhere to APA Style when constructing this assignment, including in-text citations and references for all sources that are used. Please note that no abstract is needed.

The purpose of this project is to observe and identify lighting elements utilized in an interior environment. You must discuss how each element is used in the specific applications, compare the effects of these on the appearance of a sample, and comment on the implications for interior designers and their work for clients.

OBSERVATION AND ANALYSIS
Each student will find an image of different spaces that include at least one example of commercial (educational / corporate office / restaurant / hospitality / retail) and one example of residential (home / apartment / sorority or fraternity house).

Part 1

• Identify the different types of luminaires and lighting layers seen in the image selected. The space may include more than one of the items on the list below; for example, you may find an image of an office with a luminaire that provides direct, ambient light as well as a second luminaire that provides task lighting. Or there may be only one luminaire/lighting layer visible.
• Choose a sample of interior finish material and photograph the sample in five different spaces. Photograph the sample in four different spaces within your home – Living Area, Kitchen, Bathroom, and Sleeping Area – with a fifth photograph taken outside. Identify the luminaire, lighting layer, and lamp installed, or the quality of outdoor lighting (sunny, cloudy, etc.)
• A narrative (summarized in one paragraph) must accompany each photograph in Parts 1 and 2. For images of the overall space, the narrative should explain how the lighting component was used in the space.For images of the sample seen under different lamps, the narrative should explain color change (if any) in the image compared to the actual sample, and how this will influence color selection for a specific project.

Part 2

The sample could be fabric, plastic laminate, carpeting, tile, granite, or other item. The goal is to notice the change in the appearance of the sample when seen under different lighting conditions. Make sure the sample is clearly visible so any changes in appearance can be seen in the photograph.

Part 3

• A narrative (summarized in one paragraph) must accompany each photograph in Parts 1 and 2. For images of the overall space, the narrative should explain how the lighting component was used in the space. For images of the sample seen under different lamps, the narrative should explain color change (if any) in the image compared to the actual sample, and how this will influence color selection for a specific project.·
• All documents and images are included
  in a digital document displaying the required elements.

Week of the experiment: 4

OBJECTIVE (3 points):

The objective of this lab is to gain a more complete conceptual understanding of 2-D kinematics. We will explore the concepts of position, distance, velocity, and displacement through the use of the motion data gathered with a cart simulation. This will be used to extrapolate and calculate graphs and equations for the positions and velocity.

EXPERIMENTAL DATA (3 points):

PART 1: Object moving away from the motion sensor.

Table 1

Run #	Time interval, (s)	Position (m) [Initial & End]	Distance (m)	Displacement (m)	Speed (m/s)	Velocity (m/s)
1a	8.187 to 9.843	0.85 to 1.70	0.850	0.850	0.513	0.513
1b	7.09 to 8.06	0.28 to 0.78	0.500	0.500	0.515	0.515
1c	9.281 to 10.06	1.41 to 1.81	0.400	0.400	0.513	0.513

Table 2

Run #	Slope (m/s)	Y-intercept (m)	Name of Physics quantity (i.e. position, distance, etc.)
1	0.5160 +/- 0.0001203	-3.376 +/- 0.001031	Position
2	1.645 +/- 0.0001288	-1.982 +/-0.0002569	Position

Using equation 5, substitute in your slope and y-intercept to create an equation that describes the motion of the cart:

Run 1:y=0.5160x -3.376

Run 2: y=-0.00260x+0.5383

Table 3. (Use Logger Pro for statistical data to calculate avg. velocity and uncertainty)

Run #	Average Velocity, (m/s)	Standard deviation (m)
1	0.5162	0.002718
2	1.661	0.002959

PART 2: Object moving toward the motion sensor.

Table 4

Run #	Slope (m/s)	Y-intercept (m/s)	Name of Physics quantity (i.e. position, distance, speed, etc.)
3	-.8242, +/- .0001528	3.108, +/- .0004052	Position
4	-1.235 +/- .0002127	7.175 +/- .001075	Position

Using equation 5, substitute in your slope and y-intercept to create an equation that describes the motion of the cart:

Run 3: Position = -.8242t + 3.108

Run 4: Position = -1.235t + 7.175

Table 5. (Use Logger Pro for statistical data to calculate avg. velocity and uncertainty)

Run #	Average Velocity, (m/s)	Standard deviation (m/s)
3	-.8370 +/- .002111	.002778
4	-1.275 +/- .01070	.003428

PART 3. Matching position vs. time graphs

Run 5: Include in the Discussion the methods used to reproduce the graph.

Insert the best match for the position vs time graph.

DATA ANALYSIS (10 points): the section includes equations, calculations and error analysis if required. Be sure all equations are present!

PART 1 & 2:

Show the equation with the plugged in numbers you used in run 1 to calculate the cart’s:

·Distance: xFinal – xInitial = Distance traveled

1.70 m – .85 m = .85 m

(

·Displacement: xFinal – xInitial = Displacement

1.70 m – .85 m = .85 m

·Average speed: Displacement / Time = Average speed

.85 m / (9.843 – 8.187) s = .513 m/s

·Average velocity: Displacement / Time = Average velocity

.85 m / (9.843 – 8.187) s = .513 m/s

For run 2 and 4, calculate the percent difference between the average velocity from the slope of the x(t) graph and mean velocity from v(t) graph.

Let x=slope of x(t) & v=mean velocity of v(t)

% Difference = 100 * |x+v|/((x+v)/2)

Run #2: 100 * |.516+.5162|/((.516+.5162)/2) = 0.0387%

Run #4: 100 * |-1.235+-1.275|/((-1.235+-1.275)/2) = 3.187%

PART 4:

Run 6: Show calculation (use eqn. 5) used to make a prediction for the final position of the cart after 2 s from the moment it crosses the 80 cm mark on the track.

x(t) = x0 + v(t-t0)

x(t)= (80cm) + (-20m/s)(2s-0s)

x(t)=80cm +(-40cm)

x(t)=40cm

RESULTS (3 points):

Table 6. PARTS 1 & 2 (Report results below with 4 significant figures)

	Average velocity (m/s)
Run #	Position vs. Time graph	Velocity vs. Time graph	% Difference
1	.516	.5162	0.0387%
2	1.645	1.661	0.9679%
3	-.8242	-.8370	1.541%
4	-1.235	-1.275	3.187%

Table 7. PARTS 1 & 2 Equations of motion

Run #	Substituted Equations of Motion (equation 5)
1	x(t) = x0 + .5162(t-t0)
2	x(t) = x0 + 1.661(t-t0)
3	x(t) = x0 + -.8370(t-t0)
4	x(t) = x0 + -1.275(t-t0)

Table 8. PART 4 final position of cart after 2 s

Predicted position:	Experimental position:
40.0cm	40.7cm

DISCUSSION AND CONCLUSION (10 points):