The ciphers in the earlier problems are called substitution ciphers because of the substitution technique used. A cipher is an algorithm used for encryption and decryption. Generally, there are two related algorithms, one for encryption and one for decryption. A substitution cipher is an algorithm that encrypts a message by replacing each character in the plaintext with another character. The receiver reverses the substitution to decrypt the message. A key typically consists of the information needed to carry out the encryption (or decryption) process. In a shift substitution cipher, a key is typically a number. For example, the key in a Caesar cipher is the number of places down the alphabet that you must shift.

a. What is the key for the Caesar cipher in the earlier problem?

b. What is the key for the ROT13 cipher?

c. Substitution ciphers like ROT13 and Caesar ciphers are not very secure. That is, it is not very difficult to decrypt the ciphertext even if you do not know the key. Describe some general strategies you think could be used to break substitution ciphers like those above.

d. Create your own substitution cipher. Encrypt a brief message. Then exchange encrypted messages with a classmate and try to decrypt each other's messages. Discuss and resolve any difficulties in the decryption process.

A standardized identification number system is used for books around the world. Each book is assigned a unique code-an International Standard Book Number (ISBN). These codes allow books to be more accurately identified and more effectively organized, especially with computerized recordkeeping. From 1970 to 2007, ISBNs had 10 digits. Since January 1, 2007, ISBNs contain 13 digits, so the numbering system is sometimes called ISBN-13. In this task and in Extensions Task 14, you will consider the older ISBN system. See Extensions Task 15 for an analysis of the ISBN-13 system.

Consider ISBN 0-380-00832-7. A 0 or 1 in the first position indicates that a book was published in an English-speaking country. The next block of digits identifies the publisher. The third block is assigned by the publisher from among the block of numbers the Library of Congress gives the publisher and it identifies the specific book. These first three blocks of digits can be of different lengths, depending on the publisher of the book. The last digit is a check digit. Here is how the check digit works.

There are ten digits in an ISBN: abcdefghij.

Compute 10a + 9b + 8c + 7d + 6e + 5f + 4g + 3h + 2i + j.

If the number is valid, this sum will be equivalent to 0 mod 11.

a. Verify that the ISBN given above is valid.

b. Consider this code: 0-88385-423-0. Is this a valid ISBN? Enter the number into the Search feature of a bookseller's Web site. What happens? Calculate to see if the ISBN is valid.

Rewrite each expression in standard polynomial form.

a. 3x(4 - 2x) + 8(x$^2$ - 5x)

b. (x - 5)(4x$^2$ + 7x - 9)

c. (5x$^3$ - 7x$^2$ + 5) - (x$^3$ +10x - 12)

d. $\frac{4 x^3-6 x^2+2 x}{2 x}$

Solve each equation.

a. 2t + 4(5t + 9) = 3

b. 8y(y - 2) + 4(6y) = 4(2y + 1.5y + y)

c. -0.15(p + 3) = p(0.75p -1.25) - 0.75p$^2$

d. $\frac{3}{8} x-\frac{1}{4}\left(\frac{1}{2} x+4\right)=\frac{3}{2}$

Rewrite each product as a single algebraic fraction. Then simplify the result as much as possible.

a. $\left(\frac{2 x^2}{9}\right)\left(\frac{12}{8 x}\right)$

b. $\left(\frac{3 a}{12}\right)\left(\frac{a+4}{a^2}\right)$

c. $\left(\frac{y^2-16}{y+1}\right)\left(\frac{y}{y+4}\right)$

In the diagram below, $\overleftrightarrow{A B} \| \overleftrightarrow{D E}$

a. Explain why $\triangle$ CED $\sim \triangle$ CBA.

b. Find the lengths of $\overline{A C}$ and $\overline{D E}$.

Fifty randomly selected families in a neighborhood in a large city were surveyed and each family was asked how many vehicles were registered at that address. The results of the survey are summarized in the table below.

$$\begin{array}{|c|c|c|} \hline \begin{array}{c} \text { Number of } \\ \text { Vehicles } \end{array} & \begin{array}{c} \text { Number of } \\ \text { Families } \end{array} & \begin{array}{c} \text { Percentage of } \\ \text { Families } \end{array} \\ \hline 0 & 17 & \\ \hline 1 & 23 & \\ \hline 2 & 7 & \\ \hline 3 & 2 & \\ \hline 4 & 1 & \\ \hline \end{array}$$

a. Explain what the 23 in the table indicates.

b. Use the data in the table to compute the average number of vehicles registered per address in this neighborhood.

c. Determine the percentage of families in the neighborhood that had no vehicles registered at their address. Then fill in the rest of that column of the table with percentages. Enter each percentage as a decimal number between 0 and 1. The percentage of times that a data value occurs is also called the relative frequency.

d. You can also find the average number of vehicles per address using the relative frequencies. In the formula below, f(x) is the relative frequency of the data value x. Use this formula to find the average number of vehicles registered per address.

Mean $=\sum(f(x) \cdot x)$, where x takes on all possible data values.

Your answer here should match the one you got in Part b.

e. What is the median number of vehicles registered per address in this neighborhood?

f. Which measure of center is most appropriate for these data? Explain your reasoning.

Consider the following numbers:

$$\begin{array}{lllllllll} 123 & \frac{2}{3} & \pi & -\frac{8}{4} & \sqrt[3]{8} & 31.72 & -53 & 6 \frac{3}{8} & \sqrt{7} \end{array}$$

a. Which of the numbers are whole numbers?

b. Which of the numbers are integers?

c. Which of the numbers are rational numbers?

d. Which of the numbers is irrational numbers?

e. Order the numbers from smallest to greatest.

Write each expression in a simpler equivalent exponential form using only positive exponents.

a. (3x$^6$)$^2$

b. (3$x^6$)(5x$^2$)

c. $\frac{10 x^6}{4 x^2}$

d. $\frac{-2 x^2}{8 x^6}$

e. 14(7x)$^{-1}$

f. (3x$^{-6}$)(2x$^4$)

A survey of 648 residents of Summit found that 480 of them thought the town should have a 4th of July parade.

a. What is the parameter that this survey was trying to estimate?

b. What is the point estimate of the proportion of residents of Summit who think the town should have a 4th of July parade?

c. Compute the margin of error for this survey. Then write a sentence explaining what it means in this context.

In recent years, the number of students enrolled in colleges and universities in the United States has been increasing linearly. In 2001, there were approximately 15.9 million students enrolled in U.S. colleges and universities (including community colleges). In 2012, total U.S. college and university enrollment had increased to approximately 19.8 million students.

a. Using the information above, write a function rule that can be used to estimate student enrollment in U.S. colleges and universities. Let t represent the number of years since 2000 and E(t) represents the number of students in millions.

b. What is the slope of the graph of this function? Be sure to include units of measure in your answer. What does it tell you about student enrollment in U.S. colleges and universities?

c. What is the y-intercept of the graph of this function? What does it tell you about student enrollment in U.S. colleges and universities?

d. Use your function to estimate college and university enrollment in 2015.

e. Use your function to determine the year during which the enrollment will first be greater than 23 million.

At the beginning of the school year, the 186 seniors at Sterling High School were surveyed and asked about whether they plan to go to college and if they had a summer job. Some of the results are shown in the chart below.

$$\begin{array}{|l|c|c|c|} \hline& \text { Had Summer Job } & \text { No Summer Job } & \text { Total } \\ \hline \text { College Plans } & & & \\ \hline \text { No College Plans } & 24 & & 72 \\ \hline \text { Total } & & 56 & 186 \\ \hline \end{array}$$

a. Fill in the remaining cells on a copy of the chart.

Suppose one of the seniors at Sterling High School is randomly chosen. Determine each of the following probabilities.

b. P(college plans)

c. P(had summer job)

d. P(college plans and had summer job)

e. P(college plans or had summer job)

f. What is the probability that someone who had college plans also had a summer job? (This can be written in symbols as P(had summer job | college plans). The symbol | means "given that" and restricts the group under consideration. So in this case, rather than determining the probability based on all of the students, we are only considering the 114 students who planned to go to college.)

g. P(college plans | had summer job)

Factor each of the following expressions into linear factors.

a. y$^2$ - 9

b. 3q$^2$ - 20q - 7

c. ax$^2$ - 5ax + 4a

d. 4(p - 1) + k(p - 1)

e. 6x$^2$ + 3x - 63

Often it is valuable to rewrite numbers in different but equivalent forms. Doing so may provide insight into properties of the number and it can also allow you to identify relationships between numbers.

a. Rewrite each number as a product of prime numbers.

i. 18

ii. 64

iii. 140

iv. 196

v. 320

b. Determine the greatest common divisor for each pair of numbers.

i. 18 and 6

ii. 196 and 64

iii. 64 and 320

iv. 140 and 320

v. 320 and 18

c. How can you use the prime factorization of a pair of numbers to help you determine the greatest common divisor of the numbers?