Prime number indicator: In this activity, you will design a circuit that indicates whether a number between 0 through 15 is a prime number or not. A prime number is defined as a number that can be divided only by itself and 1. By definition, the numbers 0 and 1 are not prime.

The circuit must have four binary inputs A, B, C, D, and one output, G. The output signal G is true (G = 1) if the input represents a prime number and is false (G = 0) otherwise.

a) Construct a truth table to represent this problem

b) Derive the Boolean equation for F from the truth table

c) Derive a simplified expression for F

d) Draw a logic circuit to implement your solution using AND, OR and NOT gates

e) Draw a logic circuit to implement your solution using NOR gates only

Design the circuit that gives the total number of your school number as an asynchronous counter. If 4 + 6 + 0 + 4 + 1 + 8 + 0 + 4 + 0 = 27, Mode 28 counter will be designed. FF with negative edge triggering will be used. You will definitely show the clock pulse, input and output connections of the FFs you will use. Otherwise, you will not get points because the counter will not work or it will work incorrectly. The circuit diagram will be drawn by hand

İmportant note:When solving your questions, remember to show the falling or rising edges of the signals. Do not leave cells empty in Karnough placements. Please show the groupings

In the notation, <T extends Number>, the Number class

A. Specifies both an upper and lower bound for the type T

B. Specifies a lower bound for the parameter type T

C. Specifies an upper bound for the parameter type T

D. None of the above

Neutropenia is an abnormally low number of neutrophils in the blood. Chemotherapy often reduces the number of neutrphils to a level that makes patients susceptible to fever and infections. G. Bucaneve et al. published a study of such cancer patients in the paper "Levofloxacin to Prevent Bacterial Infection in Patients With Cancer and Neutropenia" (New England Journal of Medicine, Vol. 353, No. 10, pp. 977-987). For the study, 375 patients were randomly assigned to receive a daily dose of levofloxacin, and 363 were given a placebo. In the group receiving levofloxacin, fever was present in 243 patients for the duration of neutropenia, whereas fever was experienced by 308 patients in the placebo group. (Source: Elementary Statistics, Weiss, 8th Edition)

Calculate the margin of error for a 95% confidence interval for the difference in the proportion of patients with fever between patients taking levofloxacin and those not taking anything while suffering from neutropenia. (round proportions to hundredths place, round the critical value to the hundredths place, the standard error to the thousandths place , and the margin of error to the thousandths place)

Given the language L={w| the number of a’s is greater than or equal to the number of b’s in w}

a) Using the Pumping Lemma to prove L is not a regular language.

b) Using closure property to prove L is not a regular language.

Suppose k is any natural number, k >= 0. Prove that the number of nodes in any binomial tree of height k is exactly 2^k.

C++

Create a program that checks whether a number is a prime number and displays its factors if it is not a prime number.

Console

Prime Number Checker

Please enter an integer between 1 and 5000: 5

5 is a prime number.

Try again? (y/n): y

Please enter an integer between 1 and 5000: 6

6 is NOT a prime number.

It has 4 factors: 1 2 3 6

Try again? (y/n): y

Please enter an integer between 1 and 5000: 200

200 is NOT a prime number.

It has 12 factors: 1 2 4 5 8 10 20 25 40 50 100 200

Try again? (y/n): n

Bye!

Specifications

• A prime number is divisible by two factors (1 and itself). For example, 7 is a prime number because it is only divisible by 1 and 7.
• Assume that the user will enter a valid integer.
• If the user enters an integer that’s not between 1 and 5000, the program should display an error message.
• If the number is a prime number, the program should display a message.
• If the number is not a prime number, the program should display a message. Then, it should display the number of factors for the number and a list of those factors.
• Store the factors for each number in a vector.

What is the maximum number of frames per second (round down to a whole number) that this system can display in 16,777,216 simultaneous colors at a resolution of 1920 x 1080?

L= {w belongs to {0,1}* | the number of 0's is a prime number} . Prove that the language L is not regular.