Unit 2 Review
Consider strings of lower case letters of length 4 or less. Identify the formula to calculate the number of strings of lowercase letters of length four or less, not counting the empty string.
(∑4i=0 26^i)−1
Consider all bit strings of length six or less. Identify the formula to calculate the number of bit strings of length six or less, not counting the empty string.
(∑6i=0 2^i) −1
Consider a set A with five elements. Find the number of one-to-one functions from the set A to a set with 3 elements.
0
Find the value of each of these quantities. C(5, 0)
1
Which equations arise when the steps of the Euclidean algorithm are reversed to express the greatest common divisor of each of these pairs of integers as a linear combination of these integers? (9999, 11111) (Check all that apply.)
1 = 2 · 5 − 9 1 = 247 · 1103 − 245 · 1112 1 = 2468 · 9999 − 2221 · 11111
Which equations arise when the steps of the Euclidean algorithm are reversed to express the greatest common divisor of each of these pairs of integers as a linear combination of these integers? (21, 44) (Check all that apply.)
1 = 21 - 10 · 2 1 = 21 · 21 - 10 · 44
Identify the correct step to prove that if a is an integer other than 0, then 1 divides a.
1 | a since a = 1 • a
Find the value of each of these quantities. C(5, 3)
10
An office building contains 29 floors and has 37 offices on each floor. How many offices are in the building?
1073
Identify the greatest common divisor of the following pair of integers. The greatest common divisor of 0 and 11 is
11
Convert the decimal expansion of each of the following integers to a binary expansion. 1023 The binary expansion of 1023 is
11 1111 1111
Which of these equations is produced as a step when the Euclidean algorithm is used to find the gcd of given integers? (11, 1)
11 = 11 · 1 + 0
Convert the hexadecimal expansion of each of the following integers to a binary expansion. The binary expansion of (DDFACED)16 is
1101 1101 1111 1010 1100 1110 1101
Which of these equations is produced as a step when the Euclidean algorithm is used to find the gcd of given integers? (1111, 111) (Check all that apply.)
111 = 111 · 1 + 0 1111 = 10 · 111 + 1
Consider the positive integers less than 1000. How many positive integers less than 1000 are divisible by both 7 and 11?
12
Consider a set A with five elements. Find the number of one-to-one functions from the set A to a set with 5 elements.
120
Consider all bit strings of length six or less. How many bit strings are there of length six or less, not counting the empty string?
126
A bowl contains 10 red balls and 10 blue balls. A woman selects balls at random without looking at them. What is the minimum number of balls she must select to be sure of having at least three blue balls?
13
Use a tree diagram to find the number of bit strings of length four with no three consecutive 0s. How many such bit strings are there?
13
How many subsets with an odd number of elements does a set with 18 elements have?
131072
Consider strings of five ASCII characters. Recall that there are different 128 different ASCII characters in total. The character @ is one among them. How many strings of five ASCII characters contain the character @ ("at" sign) at least once?
1321368961
A drawer contains 12 brown socks and 12 black socks, all unmatched. A man takes socks out at random in the dark. Select the least number of socks that he must take out to be sure that he has at least two black socks.
14
Identify an inverse of 7 modulo 26.
15
Consider people with three-letter initials. How many different three-letter initials are there with no letters repeated?
15600
A palindrome is a string whose reversal is identical to the string. How many bit strings of length eight are palindromes?
16
Consider the message "DO NOT PASS GO." Identify the number obtained after applying the encryption function f(p) = (p + 13) mod26 to the the number translated from the letters of the above message. (You must provide an answer before moving to the next part.)
16-1 0-1-6 2-13-5-5 19-1
Consider the message "DO NOT PASS GO." Identify the number obtained after applying the encryption function f(p) = (3p + 7) mod26 to the the number translated from the letters of the above message. (You must provide an answer before moving to the next part.)
16-23 20-23-12 0-7-9-9 25-23
How many possibilities are there for the win, place, and show (first, second, and third) positions in a horse race with 13 horses if all orders of finish are possible?
1716
A multiple-choice test contains 9 questions. There are four possible answers for each question. In how many ways can a student answer the questions on the test if the student can leave answers blank?
1953125 5^9
Consider two positive integers 29 and 55. How many positive integers between 29 and 55 are divisible by both 3 and 4?
2
Jessie and Casey are getting married. In how many ways can a photographer at their wedding arrange six people in a row from a group of 10 people, where the spouses are among these 10 people, and satisfy the given conditions? Exactly one of the spouses is in the picture. Let p be the number of ways for Jessie to be in the picture and q be the number of ways for both Jessie and Casey to be in the picture.
2(p - q)
On each of the 22 work days in a particular month, every employee of a start-up venture was sent a company communication. If a total of 4730 company communications were sent, how many employees does the company have, assuming that no staffing changes were made that month?
215
In how many ways can a set of 6 letters be selected from the English alphabet?
230230
Consider strings of eight uppercase English letters. How many strings of eight uppercase English letters are there that start with X, if no letter can be repeated?
2422728000
How many license plates can be made using either two uppercase English letters followed by five digits or three uppercase English letters followed by four digits?
243360000
Consider a set A with five elements. How many one-to-one functions are there from the set A to a set with 7 elements?
2520
Consider people with three-letter initials. What are the number of choices for the first-, second-, and third-letter initials, if none of the letters are repeated?
26, 25, and 24
A multiple-choice test contains 9 questions. There are four possible answers for each question. In how many ways can a student answer the questions on the test if the student answers every question?
262144 4 · 4 · · ·4 = 4^9
Consider all strings of three decimal digits. How many strings of three decimal digits have exactly two digits that are 4s?
27
Find the value of each of these quantities. C(8, 2)
28
How many subsets of a set with 100 elements have more than one element?
2^100 − 101
Consider bit strings of length seven. How many bit strings of length seven either begin with two 0s or end with three 1s?
2^5 + 2^4 − 2^2
A drawer contains 12 brown socks and 12 black socks, all unmatched. A man takes socks out at random in the dark. Select the least number of socks that he must take out to be sure that he has at least two socks of the same color.
3
Which memory locations are assigned by the hashing function h(k) = kmod 101 to the records of insurance company customers with the following Social Security numbers? The assigned memory location to 501338753 is
3
What sequence of pseudorandom numbers is generated using the linear congruential generator xn + 1 = (4xn + 1) mod 7 with seed x0 = 3? Enter the sequence below, starting with x0. The sequence is
3 , 6 , 4, 3
Which equations arise when the steps of the Euclidean algorithm are reversed to express the greatest common divisor of each of these pairs of integers as a linear combination of these integers? (117, 213) (Check all that apply.)
3 = 1 · 12 − 1 · 9 3 = 2 · 96 − 9 · 21 3 = 11 · 213 − 20 · 117
Consider the message "DO NOT PASS GO." Translate the letters in the above message to numbers by using their position in the alphabet.
3-14 13-14-19 15-0-18-18 6-14
Consider two positive integers 29 and 55. Which are the integers between 29 and 55 that are divisible by 3? (You must provide an answer before moving to the next part.) The integers are
30 33 36 39 42 45 48 51 54
Consider a set A with five elements. How many one-to-one functions are there from the set A to a set with 10 elements?
30240
Consider strings of eight uppercase English letters. How many strings of eight uppercase English letters are there that start with the letters BO (in that order), if letters can be repeated?
308915776
Suppose that a password for a computer system must have at least 8, but no more than 12, characters, where each character in the password is a lowercase English letter, an uppercase English letter, a digit, or one of the six special characters ∗, >, <, !, +, and =. Using the answer of part (a), determine how long, in years (round to the nearest positive integer), it takes a hacker to try every possible password assuming that it takes one nanosecond for a hacker to check each possible password. (Round the final answer to the nearest positive integer.)
314582
Consider two positive integers 29 and 55. Which are the integers between 29 and 55 that are divisible by 4? (You must provide an answer before moving to the next part.) The integers are
32 36 40 44 48 52
Find the value of each of these quantities. P(8, 3)
336
There are 21 mathematics majors and 325 computer science majors in a college. In how many ways can one representative be picked who is either a mathematics major or a computer science major?
346
Consider two positive integers 29 and 55. Which are the integers between 29 and 55 that are divisible by both 3 and 4? The integers are
36 48
Find the value of each of these quantities. P(6, 4)
360
Consider the positive integers less than 1000. How many positive integers less than 1000 have distinct digits and are even?
373
Find the number of bit strings that satisfies the given conditions. The bit strings of length 10 having at most four 1s
386
How many numbers must be selected from the set {1, 2, 3, 4, 5, 6} to guarantee that at least one pair of these numbers add up to 7?
4 numbers from the given set, since there are exactly three subsets of two integers from the given set that add up to 7
Find the value of each of these quantities. P(8, 8)
40320
In how many different orders can 8 runners finish a race if no ties are allowed?
40320
In how many ways can you create a two-element set where each element in the set is an positive integer less than 95?
4371
Consider strings of lower case letters of length 4 or less. How many strings are there of lowercase letters of length four or less, not counting the empty string?
475254
Find the number of bit strings that satisfies the given conditions. The bit strings of length 12 having exactly four 1s
495
A bowl contains 10 red balls and 10 blue balls. A woman selects balls at random without looking at them. How many balls must she select to be sure of having at least three balls of the same color?
5
Consider all strings of three decimal digits. How many strings of three decimal digits begin with an odd digit?
500
Find the total number of different permutations of the set {a, b, c, d, e, f, g}.
5040
How many bit strings of length 11 both begin and end with a 1?
512
Which memory locations are assigned by the hashing function h(k) = kmod 101 to the records of insurance company customers with the following Social Security numbers? The assigned memory location to 372201919 is
52
Every student in a discrete mathematics class is either a computer science or a mathematics major or is a joint major in these two subjects. How many students are in the class if there are 40 computer science majors (including joint majors), 23 mathematics majors (including joint majors), and 7 joint majors?
56
Which memory locations are assigned by the hashing function h(k) = kmod 101 to the records of insurance company customers with the following Social Security numbers? The assigned memory location to 104578690 is
58
Consider a set A with 10 elements. How many different functions are there from the set A to a set with three elements?
59049
Consider two positive integers 29 and 55. How many positive integers between 29 and 55 are divisible by 4?
6
Find the value of each of these quantities. P(6, 1)
6
Jessie and Casey are getting married. In how many ways can a photographer at their wedding arrange six people in a row from a group of 10 people, where the spouses are among these 10 people, and satisfy the given conditions? Jessie must be in the picture.
6 · (9 · 8 · 7 · 6 · 5)
Jessie and Casey are getting married. In how many ways can a photographer at their wedding arrange six people in a row from a group of 10 people, where the spouses are among these 10 people, and satisfy the given conditions? Both spouses must be in the picture.
6 · 5 · (1 · 1 · 8 · 7 · 6 · 5)
Consider the message "DO NOT PASS GO." Identify the number obtained after applying the encryption function f(p) = (p + 3) mod26 to the the number translated from the letters of the above message. (You must provide an answer before moving to the next part.)
6-17 16-17-22 18-3-21-21 9-17
Consider strings of eight uppercase English letters. The number of strings of eight uppercase English letters are there if no letter can be repeated is _____ × 10^10
6.3
The number of subsets with more than two elements that can be formed from a set of 99 elements is ≈ _____ × 10^29. (Enter the value in decimals. Round the answer to two decimal places.)
6.34
Which memory locations are assigned by the hashing function h(k) = kmod 101 to the records of insurance company customers with the following Social Security numbers? The assigned memory location to 432222187 is
60
Consider strings of eight uppercase English letters. How many strings of eight uppercase English letters are there that start or end with the letters BO (in that order), if letters can be repeated?
617,374,576
How many possible outcomes are there in total if the coin is flipped 6 times?
64
Find the value of each of these quantities. C(12, 10)
66
Consider strings of four lowercase letters. How many strings are there of four lowercase letters that have the letter x in them?
66351
There are 21 mathematics majors and 325 computer science majors in a college. In how many ways can two representatives be picked so that one is a mathematics major and the other is a computer science major?
6825
Suppose that a password for a computer system must have at least 8, but no more than 12, characters, where each character in the password is a lowercase English letter, an uppercase English letter, a digit, or one of the six special characters ∗, >, <, !, +, and =. How many different passwords are available for this computer system?
68^8 + 68^9 + 68^10 + 68^11 + 68^12
A committee is formed consisting of one representative from each of the 50 states in the United States, where the representative from a state is either the governor or one of the two senators from that state. The number of ways to form this committee is _____ ×10^23
7.18
Find the number of bit strings that satisfies the given conditions. The bit strings of length 8 having an equal number of 0s and 1s
70
Find the value of each of these quantities. C(8, 4)
70
Find the value of each of these quantities. P(10, 3)
720
Consider the positive integers less than 1000. How many positive integers less than 1000 are divisible by neither 7 nor 11?
779
Find the number of bit strings that satisfies the given conditions. The bit strings of length 13 having at least four 1s
7814
Find the value of each of these quantities. C(8, 7)
8
Find the value of each of these quantities. P(8, 1)
8
A particular brand of shirt comes in 14 colors, has a male version and a female version, and comes in three sizes for each sex. How many different types of this shirt are made?
84
Consider two positive integers 29 and 55. How many positive integers between 29 and 55 are divisible by 3?
9
Use Fermat's little theorem to compute the following expression. Identify the value of 3^302 mod 385 using the results of previous part (a), (b), and (c) of this question and the Chinese remainder theorem.
9
Consider a set A with 10 elements. How many different functions are there from the set A to a set with five elements?
9765625
Consider all strings of three decimal digits. How many strings of three decimal digits do not contain the same digit three times?
990
Decrypt these messages that were encrypted using the Caesar cipher. EOXH MHDQV The decrypted message of EOXH MHDQV is
BLUE JEANS
Show that among any group of five (not necessarily consecutive) integers, there are two with the same remainder when divided by 4.
Because there are four possible remainders when an integer is divided by 4, the pigeonhole principle implies that given five integers, at least two have the same remainder.
Let (xi, yi), i = 1, 2, 3, 4, 5 be a set of five distinct points with integer coordinates in the xy plane. Show that the midpoint of the line joining at least one pair of these points has integer coordinates.
By the pigeonhole principle, if we have 5 points, then at least two of these points will have the same parity (both odd or both even). The midpoint of the segment joining these two points will therefore have integer coordinates.
Let n be a positive integer. Click and drag the steps to their corresponding step numbers to show that in any set of n consecutive integers, there is exactly one divisible by n.
Continue to next card
Decrypt these messages that were encrypted using the Caesar cipher. HDW GLP VXP The decrypted message of HDW GLP VXP is
EAT DIM SUM
Consider a set A with 10 elements. There are 10^4 different functions from the set A to a set with four elements.
FALSE
Consider strings of five ASCII characters. Recall that there are different 128 different ASCII characters in total. The character @ is one among them. Identify the method to find the number of strings of five ASCII characters that contain the character @ ("at" sign) at least once.
Find the total number of ASCII strings of length five and then subtract the number of strings that do not contain the @ character.
Consider the message "DO NOT PASS GO." Translate the encrypted numbers to letters for the function f(p) = f(p + 3) mod 26. (You must provide an answer before moving to the next part.)
GR QRW SDVV JR
A palindrome is a string whose reversal is identical to the string. How many bit strings of length n are palindromes if n is even and if n is odd?
If n is even, 2^n/2; if n is odd, 2^(n + 1)/2
Find the sum and product of each of these pairs of numbers. Express your answers as base 3 expansions. (20001)3 and (1111)3
The sum of the numbers (20001)3 and (1111)3 is ( 21112 )3 and their product is ( 22221111)3
Identify the sum and product of each of these pairs of numbers. The sum of the numbers (20CBA)16 and (A02)16
The sum of the numbers (20CBA)16 and (A02)16 is ( 216BC )16 and their product is (14835D74 )16.
Find the sum and product of each of these pairs of numbers. Express your answers as base 3 expansions. (2112)3 and (12100)3
The sum of the numbers (2112)3 and (12100)3 is ( 21212 )3 and their product is (111102200)
Find the sum and product of each of these pairs of numbers. Express your answers as octal expansions. The sum of the numbers (54321)8 and (3457)8
The sum of the numbers (54321)8 and (3457)8 is ( 60000 )8 and their product is (237402537 )8
Identify the sum and product of each of these pairs of numbers. The sum of the numbers (ABCDE)16 and (1111)16
The sum of the numbers (ABCDE)16 and (1111)16 is ( ACDEF )16 and their product is (B74148BE )16.
Identify the sum and product of each of these pairs of numbers. The sum of the numbers (E0000E)16 and (BAAB)16
The sum of the numbers (E0000E)16 and (BAAB)16 is ( E0BAB9 )16 and their product is (A355AA355A )16.
Let d be a positive integer. Show that among any group of d + 1 (not necessarily consecutive) integers there are two with exactly the same remainder when they are divided by d.
There are d possible remainders when an integer is divided by d. By the pigeonhole principle, if we have d + 1 integers, then at least two must have the same remainder when divided by d.
A group contains n men and n women. Identify the steps used to find the number of ways to arrange n men and n women in a row if the men and women alternate? (Check all that apply.)
There are n men and n women, and all of the P(n, n) = n! arrangements are allowed for both men and women. There are exactly two possibilities: either the row starts with a man and ends with a woman, or it starts with a woman and ends with a man. By the product rule, there are n! · n! · 2 = 2(n!)22(n!)2 ways.
Identify the correct statement about the given integers. 17, 28, 19, 23
These integers are pairwise relatively prime because no two have a common prime factor. The integers 17, 28, 19, and 23 are pairwise relatively prime because 17, 19, and 23 are prime and 28 = 7 · 22. Hence, no pair of these numbers have a common prime factor
There are 30 students in a class. Choose the statement that best explains why at least two students have last names that begin with the same letter.
Using the pigeonhole principle, in any class of 30 students there must be at least two students who have last names that begin with the same letter since there are only 26 letters in the alphabet.
Consider a set of six classes, each meeting regularly once a week on a particular day of the week. Choose the statement that best explains why there must be at least two classes that meet on the same day, assuming that no classes are held on weekends.
Using the pigeonhole principle, in any set of six classes there must be at least two classes that meet on the same day because there are only five weekdays for each class to meet on.
Some airline tickets have a 15-digit identification number a1a2...a15, where a15 is a check digit that equals a1a2...a14 mod 7. Determine whether the given 15-digit number is a valid airline ticket identification number. 000122347322871 The 15-digit number 000122347322871 is
Valid
Some airline tickets have a 15-digit identification number a1a2...a15, where a15 is a check digit that equals a1a2...a14 mod 7. Determine whether the given 15-digit number is a valid airline ticket identification number. 101333341789013 The 15-digit number 101333341789013 is
Valid
Some airline tickets have a 15-digit identification number a1a2...a15, where a15 is a check digit that equals a1a2...a14 mod 7. Determine whether the given 15-digit number is a valid airline ticket identification number. 113273438882531 The 15-digit number 113273438882531 is
Valid
Some airline tickets have a 15-digit identification number a1a2...a15, where a15 is a check digit that equals a1a2...a14 mod 7. Which errors in a single digit of a 15-digit airline ticket identification number can be detected if a digit x in the nth column from the right is incorrectly replaced with a digit y?
We can detect all the errors except when the difference between x and y is 7.
List all the permutations of {a, b, c}.
abc, acb, bac, bca, cab, cba
Suppose that a password for a computer system must have at least 8, but no more than 12, characters, where each character in the password is a lowercase English letter, an uppercase English letter, a digit, or one of the six special characters ∗, >, <, !, +, and =. Find the number of passwords that contain at least one occurrence of at least one of the six special characters if p is the number of passwords that are available for this computer system and q is the number of passwords that do not contain any of the special characters.
p - q
Consider the positive integers less than 1000. Which rule must be used to find the number of positive integers less than 1000 that are divisible by 7?
the division rule of counting
Consider bit strings of length seven. Which of the following must be used to find the number of bit strings of length seven that either begin with two 0s or end with three 1s?
the inclusion-exclusion principle the product rule
Consider the positive integers less than 1000. Identify the rules used to find the number of positive integers less than 1000 that are divisible by exactly one of 7 and 11. (Check all that apply.)
the principle of inclusion-exclusion for sets the division rule
Consider the positive integers less than 1000. Identify the rules used to find the number of positive integers less than 1000 that are divisible by either 7 or 11. (Check all that apply.)
the principle of inclusion-exclusion for sets the division rule for counting
A committee is formed consisting of one representative from each of the 50 states in the United States, where the representative from a state is either the governor or one of the two senators from that state. Which rule must be used to find the number of ways to form this committee?
the product rule
An office building contains 29 floors and has 37 offices on each floor. Which rule must be used to find the total number of offices in the building?
the product rule
Consider a set A with 10 elements. Which rule must be used to find the number of functions from the set A to a set with two elements?
the product rule
Consider strings of eight uppercase English letters. Which rule must be used to find the number of strings of eight uppercase English letters that are there if the letters can be repeated?
the product rule
Consider strings of eight uppercase English letters. Which rule must be used to find the number of strings of eight uppercase English letters that start and end with X, if letters can be repeated?
the product rule
Consider strings of eight uppercase English letters. Which rule must be used to find the number of strings of eight uppercase English letters that start and end with the letters BO (in that order), if letters can be repeated?
the product rule
Consider strings of eight uppercase English letters. Which rule must be used to find the number of strings of eight uppercase English letters that start with X, if letters can be repeated?
the product rule
There are 21 mathematics majors and 325 computer science majors in a college. Which rule must be used to find out the number of ways that two representatives can be picked so that one is a mathematics major and the other is a computer science major?
the product rule
Consider strings of four decimal digits. Which rule must be used to find the number of strings of four decimal digits that have exactly three digits that are 9s? How many strings of four decimal digits have exactly three digits that are 9s?
the product rule 36
Consider strings of four decimal digits. Which rule must be used to find the number of strings of four decimal digits that end with an even digit? How many strings of four decimal digits end with an even digit?
the product rule 5000
Consider the positive integers less than 1000. Identify the rules used to find the number of positive integers less than 1000 that have distinct digits. (Check all that apply.)
the product rule the sum rule
Consider the positive integers less than 1000. Which of the following rules is used to find the number of positive integers less than 1000 that are divisible by 7 but not by 11? (Check all that apply.)
the subtraction rule the division rule
There are 21 mathematics majors and 325 computer science majors in a college. Which rule must be used to find out the number of ways one representative can be picked who is either a mathematics major or a computer science major?
the sum rule
Consider all bit strings of length six or less. Identify the rules used to calculate the number of bit strings of length six or less, not counting the empty string. (Check all that apply.)
the sum rule the product rule
Consider strings of lower case letters of length 4 or less. Identify the rules used to calculate the number of strings of lowercase letters of length four or less, not counting the empty string. (Check all that apply.)
the sum rule the product rule
Consider strings of lower case letters of length 4 or less. Identify the formula to calculate the number of strings of lowercase letters of length four or less.
∑4i=0 26^i
Which equations arise when the steps of the Euclidean algorithm are reversed to express the greatest common divisor of each of these pairs of integers as a linear combination of these integers? (0, 224)
224 = s · 0 + 1 · 224 for some integer s
Convert the binary expansion of each of the following integers to a decimal expansion. The decimal expansion of (1 1001)2 is
25
Find the sum and product of each of these pairs of numbers. Express your answers as octal expansions. The sum of the numbers (6001)8 and (272)8
The sum of the numbers (6001)8 and (272)8 is ( 6273 )8 and their product is ( 2134272)8.
Find the sum and product of each of these pairs of numbers. Express your answers as octal expansions. The sum of the numbers (763)8 and (144)8
The sum of the numbers (763)8 and (144)8 is ( 1127 )8 and their product is ( 141354 )8.
Evaluate these quantities. -101 mod 13 =
-101 mod 13 =
Which of these equations is produced as a step when the Euclidean algorithm is used to find the gcd of given integers? (13671, 1470) (Check all that apply.)
13671 = 9 · 1470 + 441 441 = 3 · 147 + 0 1470 = 3 · 441 + 147
List five integers that are congruent to 4 modulo 10:
4, 24, 44, 64, and 84
Convert the hexadecimal expansion of each of the following integers to a binary expansion. The binary expansion of (80E)16 is
1000 0000 1110
Identify the quotient and remainder when -1 is divided by 2.
(-1,1)
Identify the quotient and remainder when -111 is divided by 11.
(-11,10)
Identify the quotient and remainder when 0 is divided by 19.
(0,0)
Identify the quotient and remainder when 2 is divided by 5.
(0,2)
Identify the quotient and remainder when 19 is divided by 7.
(2,5)
Identify the quotient and remainder when 789 is divided by 23.
(34,7)
Identify the quotient and remainder when 5 is divided by 1.
(5,0)
Identify the quotient and remainder when 1065 is divided by 13.
(81,12)
Which of these equations is produced as a step when the Euclidean algorithm is used to find the gcd of given integers? (101, 100) (Check all that apply.)
101 = 1 · 100 + 1 100 = 100 · 1 + 0
Evaluate these quantities. -21 mod 2=
-21 mod 2= 1
Identify the integers that are congruent to 5 modulo 15. (Check all that apply.)
-25 65
Identify the greatest common divisor of the following pair of integers. 2^2 · 7^1 and 5^3 · 13^1
1
Convert the hexadecimal expansion of each of the following integers to a binary expansion. The binary expansion of (135AA)16 is
1 0011 0101 1010 1010
Find the sum and product of each of these pairs of numbers. Express your answers as binary expansions. The sum of the numbers (1110 1111)2 and (1011 1110)2 is _______________ and their product is _____________
1. 1 1010 1101 2. 1011 0001 0110 0010
Find the sum and product of each of these pairs of numbers. Express your answers as binary expansions. The sum of the numbers (100 0111)2 and (111 1000)2 is______________ And Their product is _______________-.
1. 1011 1111 2. 10 0001 0100 1000
The statement "if a | bc, where a, b, and c are positive integers and a ≠ 0, then a | b or a | c" is Identify the correct statement to justify your answer.
1. False 2. Using the counterexample 4| (2*2) but 4 does not divide 2, the given statement is false.
Evaluate these quantities. 144 mod 7 =
144 mod 7 = 4
Find the prime factorization of each of these integers, and use each factorization to answer the questions posed. The greatest prime factor of 289 is
17
Identify the greatest common divisor of the following pair of integers. 18 and 18^18
18
Find the least common multiple of each of these pair of integers. 18, 18^18
18^18
Evaluate these quantities. 199 mod 19 =
199 mod 19 = 9
Which equations arise when the steps of the Euclidean algorithm are reversed to express the greatest common divisor of each of these pairs of integers as a linear combination of these integers? (3454, 4666) (Check all that apply.)
2 = 58 − 14 · 4 2 = 44 · 120 − 29 · 182 2 = 293 · 3454 − 835 · 1212
Identify the greatest common divisor of the following pair of integers. 2 · 3 · 7 · 11 and 2 · 3 · 7 · 11
2*3*7*11
Which of these equations is produced as a step when the Euclidean algorithm is used to find the gcd of given integers? (123, 277) (Check all that apply.)
277 = 2 · 123 + 31 123 = 3 · 31 + 30 30 = 30 · 1 + 0 31 = 1 · 30 + 1
Find the prime factorization of each of these integers, and use each factorization to answer the questions posed. The smallest prime factor of 899 is
29
Identify the integers that are prime. (Check all that apply.)
29 71 97
Identify the greatest common divisor of the following pair of integers. 2^5 · 3^1 · 5^2 and 2^3 · 3^2 · 5^3
2^3 · 3^1 · 5^2
Find the least common multiple of each of these pair of integers. 2^3 · 3^4 · 5^5 and 2^1 · 3^2 · 5^2
2^3 · 3^4 · 5^5
Identify the least common multiple of two integers if their product is 2^7⋅3^8⋅5^2⋅7^11. and their greatest common divisor is 2^3⋅3^4⋅5
2^4 · 3^4 · 5 · 7^11
Find the least common multiple of each of these pair of integers. 2^5 · 7^3 and 5^4 · 13^2
2^5 · 5^4 · 7^3 · 13^2
Find the prime factorization of each of these integers, and use each factorization to answer the questions posed. The greatest prime factor of 15 is
5
Find the prime factorization of each of these integers, and use each factorization to answer the questions posed. The smallest prime factor of 85 is
5
Identify the numbers that are divisible by 17. (Check all that apply.)
68 357 A number is divisible if the given divisor divides the number with zero remainder.
Convert the binary expansion of each of the following integers to a decimal expansion. The decimal expansion of (10 1011 0101)2 is
693
factorization to answer the questions posed. The greatest prime factor of 2401 is
7
Let m be a positive integer. Show that a ≡ b (mod m) if a mod m = bmod m. Drag the necessary statements and drop them into the appropriate blank to build your proof.
Proof Method: Direct proof Proof's assumption: a mod m= b mod m Conclusion: a= b (mod m)
Convert the binary expansion of each of the following integers to a hexadecimal expansion. The hexadecimal notation of (0111 0111 0111 0111)2 is
7777
Convert the binary expansion of each of the following integers to a decimal expansion. The decimal expansion of (11 0111 1110)2 is
894
Convert the binary expansion of each of the following integers to a hexadecimal expansion. The hexadecimal notation of (1001 1001 1001 1001)2 is
9999
Convert the binary expansion of each of the following integers to a hexadecimal expansion. The hexadecimal notation of (1010 1010 1010)2 is
AAA
Identify the correct steps involved in proving that am + 1 is composite if a and m are integers greater than 1 and m is odd. (Check all that apply.)
As m is odd, we can write am + 1 = (a + 1)(am - 1 - am - 2 + am - 3 - am - 4 + ... + 1). As both a and m are greater than 1, we have 1 < a + 1 < am + 1. Thus, a + 1 is a proper factor of am + 1. Thus, we can express am + 1 as a product of two proper factors; so, am + 1 is composite.
Find the least common multiple of each of these pair of integers. 0 and 5
Does Not Exist
Convert the binary expansion of each of the following integers to a hexadecimal expansion. The hexadecimal notation of (1111 0110)2 is
F6
Find the least common multiple of each of these pair of integers. The least common multiple of 2 · 3 · 5 · 11 and 2 · 3 · 5 · 11 is 1.
FALSE
Find the sum and product of each of these pairs of numbers. Express your answers as binary expansions. The sum of the numbers (10 1010 1010)2 and (1 1111 0010)2 is_____________ and their product is ____________
The sum of the numbers (10 1010 1010)2 and (1 1111 0010)2 is 1. ( 10010011100 )2 and their product is 2. ( 1010010111010110100
Which equations arise when the steps of the Euclidean algorithm are reversed to express the greatest common divisor of each of these pairs of integers as a linear combination of these integers? (9, 10) The equation is 1 = (-1)
The equation is 1 = (-1) · 9 + 1 · 10 .
Which equations arise when the steps of the Euclidean algorithm are reversed to express the greatest common divisor of each of these pairs of integers as a linear combination of these integers? (36, 48) The equation is
The equation is 12 = (-1) · 36 + 48
Identify the correct statement about the given integers. 19, 41, 49, 64
The integers are pairwise relatively prime because no two of them have a common prime factor.
Find the sum and product of each of these pairs of numbers. Express your answers as octal expansions. The sum of the numbers (1111)8 and (776)8
The sum of the numbers (1111)8 and (776)8 is ( 2107 )8 and their product is ( 1106556 )8.
Find the sum and product of each of these pairs of numbers. Express your answers as binary expansions. The sum of the numbers (10 0000 0001)2 and (100 0000 0000)2 is
The sum of the numbers (10 0000 0001)2 and (100 0000 0000)2 is 1. ( 11000000001 )2 and their product is 2. ( 10000000010000000000 )
Identify the correct statement about the given integers. 21, 34, 55
These are pairwise relatively prime because no two of these integers share a prime factor.
Identify the correct statement about the given integers. 14, 25, 85
These integers are not pairwise relatively prime since two of them have a common prime factor 5.
Identify the correct step to prove that if a is an integer other than 0, then a divides 0.
a | 0 since 0 = a • 0
Convert the decimal expansion of each of the following integers to a binary expansion. 321 The binary expansion of 321 is
1 0100 0001
Convert the decimal expansion of each of the following integers to a binary expansion. 100632 The binary expansion of 100632 is
1 1000 1001 0001 1000
Which equations arise when the steps of the Euclidean algorithm are reversed to express the greatest common divisor of each of these pairs of integers as a linear combination of these integers? (123, 2347) (Check all that apply.)
1 = 10 − 3 · 3 1 = 37 · 10 − 3 · 123 1 = 37 · 2347 − 706 · 123
Which equations arise when the steps of the Euclidean algorithm are reversed to express the greatest common divisor of each of these pairs of integers as a linear combination of these integers? (34, 55) (Check all that apply.)
1 = 2 · 8 - 3 · 5 1 = 13 · 21 - 8 · 34 1 = 5 · 8 - 3 · 13
Convert the hexadecimal expansion of each of the following integers to a binary expansion. The binary expansion of (ABBA)16 is
1010 1011 1011 1010
Find the prime factorization of each of these integers, and use each factorization to answer the questions posed. The greatest prime factor of 103 is
103
Which of these equations is produced as a step when the Euclidean algorithm is used to find the gcd of given integers? (11891, 2238) (Check all that apply.)
11891 = 5 · 2238 + 701 2238 = 3 · 701 + 135 701 = 5 · 135 + 26 26 = 5 · 5 + 1 135 = 5 · 26 + 5 5 = 5 · 1 + 0
Identify the positive integers that are not relatively prime to 28. (Check all that apply.)
2 7 4 The unique prime factors of 28 are 2 and 7. Therefore, all integers that are divisible by 2 or by 7 are not relatively prime to 28.
Identify the greatest common divisor of the following pair of integers. 2^2 · 3^3 · 5 · 7 · 11^2 · 13 and 2^9 · 3^8 · 11^1 · 17
2^2 · 3^3 · 11^1
Find the least common multiple of each of these pair of integers. 2^4 · 3^1 · 5 · 7 · 11^2 · 13 and 2^8 · 3^4 · 11^1 · 17^14
2^8 · 3^4 · 5 · 7 · 11^2 · 13 · 17^14
Identify the prime factorization of 10!.
2^8 · 3^4 · 5^2 · 7
Convert the binary expansion of each of the following integers to a decimal expansion. The decimal expansion of (111 1100 0001 1111)2 is
31775
Find the sum and product of each of these pairs of numbers. Express your answers as base 3 expansions. The sum of the numbers (112)3 and (210)3 is _______ and their product is ____
The sum of the numbers (112)3 and (210)3 is 1. ( 1022 )3 and their product is 2. ( 101220 )3
Find the sum and product of each of these pairs of numbers. Express your answers as base 3 expansions. (120021)3 and (2012)3
The sum of the numbers (120021)3 and (2012)3 is ( 122110 )3 and their product is (1020100022 )3
Identify the sum and product of each of these pairs of numbers. The sum of the numbers (1AE)16 and (BBD)16
The sum of the numbers (1AE)16 and (BBD)16 is ( D6B )16 and their product is ( 13B776)16.
Identify the correct steps involved in showing that if 2^n − 1 is prime, then n is prime. (Check all that apply.)
We will prove by contrapositive. Suppose n is not prime. Then, n= ab, for some integers a > 1 and b > 1. We must prove that 2ab − 1 is not prime. Consider the identity 2ab − 1 = (2a − 1) ⋅ (2a(b − 1) + 2a(b − 2) + ... + 2a + 1). The identity is valid, since we can clearly see on the right-hand side that all terms except 2ab and -1 cancel. Clearly, (2a(b − 1) + 2a(b − 2) + ... + 2a + 1) is greater than 1. Sincea > 1, the factor 2a − 1 is greater than 1. Since 2n − 1 is the product of two integers that are greater than 1, 2n − 1 is not prime.
Students, each of whom comes from one of the 50 states, must be enrolled in a university to guarantee that there are at least 100 who come from the same state. Identify which of the following statement(s) is/are correct about the minimum number of students who must be enrolled to fulfill the given condition. (Check all that apply.)
If there is no state with at least 100 students coming from that state, there can be at most 99 students coming from each state. Thus, the total number of students would be at most 50 × 99. Thus, contrapositively, if there are at least 50 × 99 + 1 = 4951 students, then there must be at least one state with at least 100 students coming from that state. By the generalized pigeonhole principle, if N students are placed into 50 boxes (with states being the boxes), then there is at least one box containing at least ⌈N50⌉⌈N50⌉ students. We want ⌈N50⌉≥100⌈N50⌉≥100 , which is equivalent to N50>99N50>99 , or N > 50 × 99. Since N is an integer, N > 50 × 99 implies N≥50×99+1=4951N≥50×99+1=4951 .
Consider seven integers selected from the first 10 positive integers. Is the conclusion in the part (a) of the question true if six integers are selected rather than seven?
NO
Some airline tickets have a 15-digit identification number a1a2...a15, where a15 is a check digit that equals a1a2...a14 mod 7. Determine whether the given 15-digit number is a valid airline ticket identification number. 007862342770445 The 15-digit number 007862342770445 is
Not Valid
Consider the message "DO NOT PASS GO." Translate the encrypted numbers to letters for the function f(p) = f(p + 13) mod 26. (You must provide an answer before moving to the next part.)
QB ABG CNFF TB
Consider the message "DO NOT PASS GO." Translate the encrypted numbers to letters for the function f(p) = f(3p + 7) mod 26.
QX UXM AHJJ ZX
Consider strings of four lowercase letters. Identify the method used to find the number of strings of four lowercase letters that have the letter x in them.
Subtract from the number of strings of length four of lowercase letters the number of strings of length four of lowercase letters without the letter x.
Decrypt these messages that were encrypted using the Caesar cipher. WHVW WRGDB The decrypted message of WHVW WRGDB is
TEST TODAY
If m and n are integers greater than or equal to 2, then the Ramsey numbers R(m, n) and R(n, m) are equal, i.e., R(m, n) = R(n, m).
TRUE
Some airline tickets have a 15-digit identification number a1a2...a15, where a15 is a check digit that equals a1a2...a14 mod 7. The accidental transposition of two consecutive digits in an airline ticket identification number can be detected using the check digit. (Assume that the difference between the transposed digits is not 7.)
TRUE
Consider strings of four decimal digits. Which rule must be used to find the number of strings of four decimal digits that do not contain the same digit twice? How many strings of four decimal digits do not contain the same digit twice?
The Product Rule 5040
