Discrete Mathematics week5
-1 is divided by 4
(-1,3) -1 = -1 *4 +3
-111 is divided by 11 (quotient, remainder)
(-11,10) -111= -11*11 +10
2 is divided by 19
(0,2) 2 = 0 *19 +2
1001 is divided by 13
(77,0)
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.
(∑^4 _i=0 26i)−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.
(∑^6 _i=0 2i)−1
Find the coefficient of x^9 in (2 - x)^19.
-94595072
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 The first choice can be made in 3 ways, since any element of the codomain can be the image of the first element of the domain. After that choice has been made, in each step choices are reduced by one. Continuing in this way and applying the product rule, we see that there are 3 ⋅ 2 ⋅ 1 ⋅ 0 ⋅ 0 one-to-one functions from a set with five elements to a set with 3 elements.
The smallest prime factor of 187 is
11
Find the number of strings of six lowercase letters from the English alphabet, which satisfies the given conditions. Six lowercase letters contain the letters a and b.
11737502
The number of subsets with more than two elements that can be formed from a set of 100 elements is ≈ _____ × 10^29. (Enter the value in decimals. Round the answer to two decimal places.)
12.68 There are 2^100 subsets of a set with 100 elements. All of them have more than two elements except the empty set, the 100 subsets consisting of one element each, and the C(100, 2) = 12.68 subsets with two elements.
How many permutations of the letters ABCDEFGH satisfies the given conditions? Permutation contains the strings BA and FGH.
120
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 5 ⋅ 4 ⋅ 3 ⋅ 2 ⋅ 1 one-to-one functions from the set A to a set with 5 elements.
Find the number of bit strings that satisfies the given conditions. The bit strings of length 9 having exactly four 1s
126 9!/4!(9 - 4)! = 9!/4! ⋅ 5! = (9 ⋅ 8 ⋅ 7 ⋅ 6)/(4 ⋅ 3 ⋅ 2 ⋅ 1)
Find the number of strings of six lowercase letters from the English alphabet, which satisfies the given conditions. Six lowercase letters contain the letters a and b in consecutive positions with apreceding b, with all the letters distinct.
1275120
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 club has 26 members. How many ways are there to choose four members of the club to serve on an executive committee?
14950 C(26,4)=26!/4!(26−4)!= 26!/4!= 26 · 25 · 24 · 23/4 · 3 · 2 · 1
A palindrome is a string whose reversal is identical to the string. How many bit strings of length eight are palindromes?
16 2^(n/2)
How many positive integers between 5 and 31 are divisible by both 3 and 4?
2 An integer is divisible by both 3 and 4 if and only if it is divisible by their least common multiple, which is 12. There are two such integers between 5 and 31, i.e., ⌊31/12⌋−⌊5/12⌋=2−0=2
Consider the statement that if n is a positive integer, then (2n2)=2(n2)+n2(22n)=2(2n)+n2 .
2(n2)+n2=2n!/(n−2)! +n2 = 2n!/ (n−2)!+n2 =2n!/2!=(2n/2)
Find the number of bit strings that satisfies the given conditions. The bit strings of length 6 having an equal number of 0s and 1s
20 6!/3! (6-3)!
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 4840 company communications were sent, how many employees does the company have, assuming that no staffing changes were made that month?
220 4840/22=220
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, 226)
226 = s * 0 +1. 226 for some integer s
How many permutations of the letters ABCDEFGH satisfies the given conditions? Permutation contains the strings CAB and BED?
24
Find the smallest number of words from a dictionary we need to select to be sure that at least 100 start with the same letter
2575
How many subsets of a set with 100 elements have more than one element?
2^100 − 101
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 For D, we have p=3 because D is the fourth letter of the alphabet. Similarly, convert all the letters to numbers
Find the coefficient of x^6 y^8 in (x + y)^14
3003 14!/(8! · 6!)
There are 16 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?
341 16+325
A club has 26 members. How many ways are there to choose a president, vice president, secretary, and treasurer of the club, where no person can hold more than one office?
358800
How many positive integers less than 1000 have distinct digits and are even?
373 To compute the two-digit numbers, note that the first digit must be 1 through 9 (9 choices), and the second digit must be something different from the first digit (9 choices out of the 10 possible digits). Thus, by using the product rule, we get 9 · 9 = 81 choices in all. This approach also tells us that there are 9 · 9 · 8 = 648 three-digit numbers with distinct digits. Then, by using the sum rule, we find that there are 9 + 81 + 648 = 738 positive integers less than 1000 that have distinct digits. Now, there are five odd one-digit, 40 odd two-digit, and 320 odd three-digit numbers with distinct digits. Hence, in total there are 5 + 40 + 320 = 365 odd numbers with distinct digits. The number of even numbers with distinct digit is the complement, 738 - 365 = 373
Find the number of strings of six lowercase letters from the English alphabet, which satisfies the given conditions. Six lowercase letters contain the letters a and b, where a is somewhere to the left of bin the string, with all the letters distinct.
3825360
How many permutations of the letters ABCDEFGH satisfies the given conditions? Permutation contains the string ED.
5040
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 43 computer science majors (including joint majors), 23 mathematics majors (including joint majors), and 7 joint majors?
59 Let A1 denote the set of students who are computer science majors and A2 denote the set of students who are mathematics majors; thus, |A1| = 43 and |A2| = 23. There are 7 students with joint majors, i.e., |A1 ∩ A2| = 7. We can use the inclusion-exclusion principle to determine the number of students in the class. |A1 ∪ A2| = |A1| + |A2| - |A1 ∩ A2| |A1 ∪ A2| = 43 + 23 - 7 = 59
Consider a set A with 10 elements. How many different functions are there from the set A to a set with three elements?
59049
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) The photographer can place Jessie in any of the six positions, and then place Casey in any of the five remaining positions. 1 · 1 · 8 · 7 · 6 · 5 = 1680 combinations times 6 · 5 ways to compute each combination. Therefore, the answer is 6 · 5 · (1 · 1 · 8 · 7 · 6 · 5).
How many functions are there from a set with four elements to a set with five elements? How many of the functions from a set with four elements to a set with five elements are one-to-one? How many of the functions from a set with four elements to a set with five elements are onto?
625 120 0
Find the number of strings of six lowercase letters from the English alphabet, which satisfies the given conditions Six lowercase letters contain the letter a.
64775151
Suppose that a password for a computer system must have at least 10, but no more than 14, 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?
68^10 + 68^11 + 68^12 + 68^13 + 68^14
The greatest common divisor of 0 and 7 is
7
A coin is flipped where, each flip comes up as either heads or tails. How many possible outcomes contain exactly two heads if the coin is flipped 13 times?
78 13!/2!(13 - 2)!
Find the number of bit strings that satisfies the given conditions. The bit strings of length 13 having at least four 1s
7814
A coin is flipped where, each flip comes up as either heads or tails. How many possible outcomes are there in total if the coin is flipped 13 times?
8192 Each flip can be either heads or tails. So, there are 2^13 total outcomes.
How many positive integers between 5 and 31 are divisible by 3?
9 Both 5 and 31 are not divisible by 3. So, ⌊31/3⌋=10 31/3=10 integers less than 31 are divisible by 3 and ⌊5/3⌋=15/3=1 integer less than 5 is divisible by 3. Thus, there are 10 - 1 = 9 integers between 5 and 31 that are divisible by 3.
Consider a set A with 10 elements. How many different functions are there from the set A to a set with five elements?
9765625 There are 5 elements in the codomain. So, there are 5 choices for a function value for each of the 10 elements in the domain. Then the total number of functions from the set A to a set with five elements is obtained using the product rule. The total number of such functions = 5 ⋅ 5 ⋅ 5 ⋅ 5 ⋅ 5 ⋅ 5 ⋅ 5 ⋅ 5 ⋅ 5 ⋅ 5 = 5^10 = 9,765,625.
Decrypt these messages that were encrypted using the Caesar cipher. EOXH MHDQV The decrypted message of EOXH MHDQV is
Blue Jeans
Consider a set A with 10 elements. There are 104 different functions from the set A to a set with four elements. True or False
False
Determine the number of functions from the set {1, 2, . . . , n} to the set {0, 1} that satisfy the indicated additional conditions for n ≥ 3. Match each additional condition with the corresponding number of functions by dragging a number from the right to the condition on the left.
Functions that are one-to-one : 0 Functions that assign 0 to both 1 and n : 2^n-2 Functions that assign 1 to exactly one of the positive integers less than n: 2(n-1)
Consider the message "DO NOT PASS GO." Translate the encrypted numbers to letters for the function f(p) = f(p + 3) mod 26.
GR QRW SDVV JR First, replace the letters in the message with numbers. This produces3 14 13 14 19 15 0 18 18 6 14.Now, replace each of these numbers p by f (p) = (p + 3) mod 26. This gives6 17 16 17 22 18 3 21 21 9 17.Translating this back to letters produces the encrypted message "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^(2n + 1)
Consider the statement "If n and k are integers with 1≤k≤n1≤k≤n , then k(nk)=n(n−1k−1).k(kn)=n(k−1n−1). "
Step 1: For the left-hand side, choose the k-set Step 2: This can be done in C(n,k) ways Step 3: Choose one of the k elements in this subset to be the distinguished element Step 4: This can be done in k ways Step 5: For the right-hand side, choose the distinguished element out of the entire n-set Step 6: This can be done in n ways Step 7: Choose the remaining k-1 elements of the subset from the remaining n-1 elements of the set Step 8: This can be done in C(n-1, k-1) ways
Click and drag the statements in the right column to left to prove the statement that (nk)≤2n(kn)≤2n for all positive integers n and all integers k with 0≤k≤n0≤k≤n
Step 1: Let n be a nonnegative integer, then C(n,0) + C(n,1) +...+C(n,n)=2^n Step 2: So, each one of them is no bigger than this sum
19, 41 , 49, 64
The integers are pairwise relatively prime because no two of them have a common prime factor. 19=19*1 41 is prime 49=7*7 64=2*2*2*2*2*2
Consider the statement that if n is a positive integer, then (2n2)=2(n2)+n2(22n)=2(2n)+n2 .
The left-hand side counts ways to do so, since simply choosing 2 people from 2n people. The right-hand side counts the number of ways to do this (by the sum rule). There are C(n, 2) ways to choose two men or two women and n⋅ n ways to choose one of each sex. To choose 2 people from a set of n men and n women, either choose 2 men or 2 women or one of each sex.
Identify the rules used to find the number of positive integers less than 1000 that have distinct digits.
The product rule the sum rule
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?
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.
a divides 0
a | 0 since 0=a*0 If a and b are integers with a does not = 0, we say that a divides b if there is an integer c such that b=ac. a | 0 since 0=a *0
Suppose that a password for a computer system must have at least 10, but no more than 14, 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
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
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.
the principle of inclusion-exclusion for sets the division rule for counting
Identify the rules used to find the number of positive integers less than 1000 that are divisible by either 7 or 11.
the principle of inclusion-exclusion for sets the division rule for counting
An office building contains 23 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
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 The product rule must be used in this case since there are 10 ways to choose the first digit, nine ways to choose the second digit, eight ways to choose the third digit, and seven ways to choose the fourth digit. Therefore, the number strings of four decimal digits do not contain the same digit twice is 10 · 9 · 8 · 7
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
the product rule the sum rule
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?
the subtraction rule the division 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
the sum rule the product 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
the sum rule the product rule We use the sum rule and the product rule, adding the number of bit strings of each length up to six by using the formula for the sum of a geometric progression.
1 divides a
1 | a since a=1*a. If a and b are integers with a does not = 0, we say that a divides b if there is an integer c such that b=ac. 1 | a since a =1*a
Which of these equations is produced as a step when the Euclidean algorithm is used to find the gcd of given integers? (14038,1529)
14038=9*1529+277 1529=5*277+144 277=1*144+133 133=12*11+1 144=1*133+11 11=11*1+0
Which of these equations is produced as a step when the Euclidean algorithm is used to find the gcd of given integers? (14039,1529)
14039= 9*1529 +278 1529=5*278+139 278=2*139+0
19 is divided by 7 (quotient, remainder)
(2,5) When 19 is divided by 7, 7 does into 19 only 2 times, creating 14 and a remainder of 5. 19 = 2*7 +5. Therefore, the quotient and remainder are 2 and 5 respectively.
The smallest prime factor 713 is
23
Identify the value of 3^302 mod 385 using the results of previous part (a), (b), and (c) of this question and the cheese remainder theorem
9
Which of these equations is produced as a step when the Euclidean algorithm is used to find the gcd of given integers? (91,90)
91=1*90+1 90=90*1+0
The greatest prime factor of 97 is
97 97 is a prime number
find the least common multiple of 19, 19^19
19^19
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)
1= 2 * 8 - 3 * 5 1= 13 *21 - 8* 34 1 = 5 * 8 -3 * 13
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)
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)
1=21*21-10*44 1=21-10*2
Identify the positive integers that are not relatively prime to 24
2,3,6 The unique prime factors of 24 are 2 and 3. Therefore, all integers that are divisible by 2 or by 3 are not relatively prime to 24.
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 A particular brand has 14 colors of shirts, 2 versions for both male and female, and comes in 3 sizes for each sex; the product rule applies and yields that there are 14 · 2 · 3 types of shirts that can be made.
Find the prime factorization of 10!
10!=10*9*8*7*6*5*4*3*2*1 2,3,5, and 7 are primes 10=2*5 9=3*3 8=2*2*2 6=2*3 4=2*2 10!= (2*5)*(3*3)*(2*2*2)*7*(2*3)*5*(2*2)*3*2 10!=2^8 * 3^4 * 5^2 * 7
How many bit strings of length 12 both begin and end with a 1?
1024 A bit string is determined by choosing the bits in the string, one after the another; so, the product rule applies. We need to count the number of bit strings of length 12 in which both the first bit and the last bit are equal to 1. Thus, we have two choices each (0 or 1) for the remaining 12 - 2 = 10 bits; so, using the product rule, there are 210 choices. Therefore, there are 210 = 1024 bit strings of length 12 that both begin and end with a 1.
Which of these equations is produced as a step when the Euclidean algorithm is used to find the gcd of given integers? (11111,1111)
1111=1111*1+0 11111=10*111+1
identify an inverse of 7 modulo 26
15 15*7 =1 (mod 26) 15*7 -1=104 is divisible by 26
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 There are 26 choices for the first initial, 25 choices for the second initial, and 24 choices for the third initial if no letter is to be repeated.
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)
2=58-14*4 2=44*120-29*182 2=293*3454-835*1212
2^2 * 3^3 * 5^5 and 2^5 * 3^2 * 5^2 Identify the greatest common divisor
2^2 * 3^2 * 5^2
Identify the correct steps involved in proving that a^ m +1 is composite if a and m are integers greater then 1 and m is odd
As m is odd, we can write a^m +1 = (a+1)(a^m-1 - a^m-2 + a^m-3 - a^m-4)+...-1). As both a and m are greater than 1, we have 1< a +1 < a ^m +1. Thus, a +1 is a proper factor of a ^m +1 Thus, we can express a^m +1 as a product of two proper factors; so, a ^m + 1 is composite
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 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.
Prove if a, b, a and d are integers where a does not =0 and b does not =0 such that a | c and b | d ten ab | cd. If ..... Then ....
If... a | c and b | d, there are integers s and t such that c=as and d=bt. Then... cd=(as)(bt)=ab(st). It follows that ab | cd.
Let m be a positive integer. Show that a = b (mod m) if a mod m = b mod m. Build your proof
Proof method: Direct Proof Proof's assumption(s): a mod m = b mod m Implication and deduction resulting from the assumptions: a = q(1) m + r and b = q(2) m +r for some integers q(1), q(2) an dan integer r with 0 less then or equal to r < m Step 1: a -b = q(1) m -q(2) m Step 2: a -b = (q(1)-q(2))m Step 3: m | (a-b) Conclusion: a = b (mod m)
Arrange the steps in the correct order to compute 3^ 302 mod 5
Step 1: 3^4 =1 (mod 5) step 2: 3^300 = (3^4)^75 =1^75 = 1(mod 5) step 3: 3^302 = 3^2 *3^300 = 9*1=9 (mod 5) step 4: 3^302 mod 5=4
Show that an inverse of a modulo m, where a is an integer and m>2 is a positive integer, does not exist if gcd(a,m)>1
Step 1: If x is an inverse of a modulo m, then by definition, ax-1=tm for some integer t Step 2: if a and m in the equation Bothe have a common divisor greater than 1, then 1 must also have this same common divisor, since 1=ax-tm step 3: This is absurd, since the only positive divisor of 1 is 1. Therefore, no such x exists.
21, 31, 55
These are pairwise relatively prime because no two of these integers share a prime factor.
14, 25, 85
These integers are not pairwise relatively prime since two of them have a common prime factor 5
Identify the integers that are congruent to 5 modulo 17
56 and -29 -29 -5 = -34. -34 is divisible by 17 56-5 = 51 is divisible by 17
789 is divided by 23 (quotient, remainder)
(34,7) 789=34* 23 +7
Find the least common multiple of each of these pair of integers
0 and 13
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 If a, ar, ar2, . . ., arn, . . . is a geometric progression, then ∑nj=0arj={arn+1−ar−1 if r≠1(n+1)a if r=1∑j=0narj=arn+1-ar-1 if r≠1(n+1)a if r=1 , where a is the initial term, r is the common ratio, and r ≠ 0. Here, a =1 and r = 2. Hence, (∑6i=02i)=26+1−12−1=27−1(∑i=062i)=26+1-12-1=27-1 . Therefore, the number of bit strings excluding the empty string = (27 - 1) - 1 = (128 - 1) - 1 = 126.
Consider people with three-letter initials. How many different three-letter initials are there with no letters repeated?
15600 26, 25, and 24 are the number of choices for the first, second, and third-letter initials, if none of the letters are repeated. Then, the number of different three-letter initials with none of the letters repeated is 26 ⋅ 25 ⋅ 24.
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.
16-1 0-1-6 2-13-5-5 19-1 Add 13, and reduce modulo 26 for 3-14 13-14-19 15-0-18-18 6-14.
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.
16-23 20-23-12 0-7-9-9 25-23 Multiply by 3 and then add 7, and reduce modulo 26 for 3-14 13-14-19 15-0-18-18 6-14.
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)
1=10-3*3 1=37*10-3*123 1=37*2347-706*123
Find the least common multiple of each of these pair of integers 2^4 * 3^3 * 5^3 and 2^2 * 3^5 *5^2
2^4 * 3^5 *5^3
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 (2^7 * 3^8 *5^2 *7^11) / (2^3 * 3^4 *5) = 2^4 *3^4 *5* 7^11
The great prime factor of 81 is
3
Which of these equations is produced as a step when the Euclidean algorithm is used to find the gcd of given integers? (165,346)
346=2*165+16 165=10*16+5 5=5*1+0 16=3*5+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? (117, 213)
3=1 * 12 -1 * 9 3 = 2 * 96 - 9 * 21 3 = 11* 213 -20 * 117
List five integers that are congruent t o4 modulo 11:
4, 26, 48, 70, and 92
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 Hence, ∑4i=026i=264+1−126−1=265−125=475,255∑i=0426i=264+1-126-1=265-125=475,255 . Therefore, the number of lowercase strings of length four or less not counting the empty string = (∑4i=026i)−1∑i=0426i-1 = 475,255 - 1 = 475,254.
There are 16 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?
5200 16 * 325
Which memory locations are assigned by the hashing function h(k) = k mod 101 to the records of insurance company customers with the following Social Security numbers? The assigned memory location to 104578690 is
58
Which of these equations is produced as a step when the Euclidean algorithm is used to find the gcd of given integers? (5,1)
5=5*1+0
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.
6-17 16-17-22 18-3-21-21 9-17 add 3, and reduce modulo 26 for 3-14 13-14-19 15-0-18-18 6-14
Identify the numbers that are divisible by 17
68 and 357
A multiple-choice test contains 11 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?
Since there are 11 tasks and four ways to do each task, the product role applies and yields that there are 4*4*4*4... = 4^11 4194304
Consider the message "DO NOT PASS GO." Translate the encrypted numbers to letters for the function f(p) = f(p + 13) mod 26.
QB ABG CNFF TB First, replace the letters in the message with numbers. This produces3 14 13 14 19 15 0 18 18 6 14.Now, replace each of these numbers p by f(p) = (p + 13) mod 26. This gives16 1 0 1 6 2 13 5 5 19 1.Translating this back to letters produces the encrypted message "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 First, replace the letters in the message with numbers. This produces3 14 13 14 19 15 0 18 18 6 14.Now, replace each of these numbers p by f(p) = (3p + 7) mod 26. This gives16 23 20 23 12 0 7 8 8 25 23.Translating this back to letters produces the encrypted message "QX UXM AHJJ ZX."
A multiple-choice test contains 11 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?
Since there are 11 tasks and five ways to do each task, the product role apples and yields that there are 5*5*5*5...= 5^11 48828125
Arrange the steps in correct order to solve the congruence 2x=7 (mod 17) using the inverse of 2 modulo 17, which is 9
Step 1: 9 is an inverse of 2 mod 17. The given equations is 2x= 7 (mod 17). Step 2: Multiplying both sides of the equation by 9, we get x =9* 7(mod 17). Step 4: since 63 mod17 =12, the solutions are all integers congruent to 12 modulo 17, such as 12, 29, and -5.
Show the steps that prove a mod m = b mod m if a = b (mod m) where m be a positive integer
Step 1: Assume that a =b (mod m) Step 2: This means that m | a -b, say a - b = mc, so that a = b+mc. Step 3: Let us compute a mod m Step 4: We know that b= qm +r for some nonnegative r less then m Step 5: We can write a = qm +r mc = (q+c) m +r STep 6: By definition, this means that r must also equal a mod m
prove that if n is an odd positive integers then n^2 = 1 ( mod 8)
Step 1: By definition of odd number, n= 2k+1 for some integer k Step 2: then n^2 = (2k +1)^2 = 4k ^2 +4k +1= 4k(k+1) +1 Step 3: Since either k or k +1 even, k(k+1) is even. Thus, 4k(k+1) is a multiple of 8 Step 4: Since n^2 -1 = 4k(k+1) is a multiple of 8, n^2 =8/ +1 , where / is a integer. Thus n^2 = 1 (mod 8) by definition.
show that if a | B and B| a, where a and b are integers, then a=b or a=-b
Step 1: If a | B and B | a, there are integers c and d such that b=ac and a = bd. Step 2: Hence, a =acd Step 3: Because a does not = 0, it follows that cd=1 Step 4: Thus, either c=d =1 or c-d=-1. Thus, a=b or a=-b
Arrange the steps in correct order to encrypt the message ATTACK using the RSA system with n = 43 · 59 and e = 13, translating each letter into integers and grouping together pairs of integers.
Step 1: Translate the letters into numbers. 0019 1900 0210 STep 2: Compute C = P13 mod 2537 for P = 19, P = 1900, and P = 210. Step 3: The results are 2299, 1317, and 2117, respectively. Step 4: The message is 2299 1317 2117.
The steps in correct order to show that if 2^m +1 is an odd prime, then m= 2^n for some nonnegative integer n
Step 1: We first prove that the polynomial identity x^m +1 = (x^k +1) (x^k(t-1)-x^k(t-2) + ... - x^k +1) is true when m=kt and t is odd Step 2: Let y=x^k and m =kt. Then the polynomial identity becomes y^t +1 = (y+1)(y^t-1 - y^t-2 +... -y+1). Clearly, this identity is true if t is odd. Thus, the original identity is true if m=kt and t is odd. Step 3: We must show that m is a power of 2 i.e., 2 is the only prime factor of m. On the contrary, suppose that t is an odd prime factor of m STep 4: Let x=2. Then, using the polynomial identity, we get 2^m +1 = (2^k +1)(2^k(t-1) -2^k(t-2) + ... -2^k +1). Step 5: As 2^k +1 > 1 and 2^m +1 is an odd prime, i.e., 2^m +1 has no proper factor greater then 1, we have 2^k +1=2m+1 Step 6: 2^k +1 =2^m +1 implies m=k and t=1. This contradicts out assumption that 2^m +1 is odd prime. Therefore, 2 is the only prime factor of m and m=2^n for some n.
Arrange the steps in the correct order to find an inverse of a modulo m for each of the following pairs of relatively prime integers using the Euclidean algorithm a=34, m =89
Step: Let 34s +89t=1, where s is the inverse of 34 modulo 89 step 2: The steps to find gcd(34,98)=1 using the Euclidean algorithm step 3: The gcd in terms of 34 and 89 is written as 1 =3-2 Step 4: s=-34 so an inverse of 34 modulo 89 is -34 which can also be written as 55
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
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. The 15-digit number 113273438882531 is ...?... Correctairline ticket identification number.
The 15-digit number 113273438882531 is... a valid ...Correctairline ticket identification number.
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.
∑^4 26i
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 , , , , ... forever.
The sequence is 3 , 6, 4, 3
There are 16 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
17, 20, 19, 23
These integers are pairwise relatively prime because no two have a common prime factor 17 is a prime number 19 is a prime number 23 is a prime number 20= 5*2*2 No pair of these numbers have a common prime factor
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 or false
True
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
Identify the correct steps involved in showing that if 2^n -1 is prime, then n is prime
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 2^ab - 1 is not prime. Consider the identity 2^ab-1 =(2^a-1) * (2^a(b-1) +2^a(b-2) +... +2^a +1). The identity is valid, since we can clearly see on the right-hand side that all terms except 2^ab and -1 cancel. Clearly, 2^a(b-1) +2^a(b-2) +... +2^a +a) is greater than 1. since a>1, the factor 2^a -1 is greater than 1 Since 2^n -1 is the product of two integers that are greater than a, 2^n -1 is not prime.
give value of i to find 11^644 mod 645
i=0 = 11^ 2 mod 645= 121 i=1 = 11^4 mod 645 = 451 i=2 = 11^8 mod 645=226 i=3 = 11^16 mod 645= 121 i=4 = 11^32 mod 645 = 451 i=5 = 11^286 mod 645 = 451 i=6 = 11^ 128 mod 645= 121 i=7 = 11^256 mod 645 = 451 i=8 = 11^512 mod 645= 226 i=9 = 11^644 mod 645 = 1
The statement "if a | bc, where a , b and c are positive integers and a does not =0, then a |b or a |c is ___T__or__F_____ This is justified by?
if a | bc, where a , b and c are positive integers and a does not =0, then a |b or a |c is ___False__ Justified Using the counterexample 4 | (2*2) but 4 does not divide 2, the given statement is false.
prove a | b and c | a , then a |c
step 1. Suppose a | B and B | c. By definition of divisibility, a | b means that b=at for some integer t ,and b | c means that c= bs for some integer s. step 2. We substitute the equation b=at into c=bs and get c=ats. step 3. By definition of divisibility, c=a(ts), with ts being an integer, implies a | c.
Arrange the steps in the correct order to find an inverse of a modulo m for each of the following pairs of relatively prime integers using the Euclidean algorithm a=200, m=1001
step 1: By using the Euclidean algorithm, 1001=5*200+1 Step 2: Let 200s +1001t =1, where s is an inverse of 200 modulo 1001 step 3: The gcd in terms of 1001 and 200 is written as 1= 1001-5 *200 step 4: s=-5 so an inverse of 200 modulo 1001 is -5
prove if a, b, and c are integers, where a does not =0 and c does not =0, such that ac | bc , then a | b
step 1: If ac | bc, then there is an integer k such that bc=ack Step 2: Hence, b=ak -> a | b
Arrange the steps in the correct order to find an inverse of a modulo m for each of the following pairs of relatively prime integers using the Euclidean algorithm a=4, m =9
step 1: The steps to find gcd(4,9) using the Euclidean algorithm are listed below, 9=2*4 +1 4=4*1 Step 2: The gcd in terms of 4 and 9 written as 1= 9-2*4 Step 3: Then boxout coefficients of 9 and 4 are 1 and -2, respectively step 4: the coefficient of 4 is -2, which is the same as 7 modulo 9 step 5: 7 is an inverse of 4 modulo 9
Arrange the steps in the correct order to find an inverse of a modulo m for each of the following pairs of relatively prime integers using the Euclidean algorithm a=2, m=17
step 1: by using the Euclidean algorithm, 17 = 8*2+1 step 2: the gcd in terms of 2 and 17 is written as 1= 17-8*2 step 3: The besot coefficients of 17 and 2 are 1 and -8, respectively step 4: the coefficient of 2 is same as 9 modulo 17 step 5: 9 is an inverse of 2 modulo 17
Arrange the steps in the correct order to solve the system of congruencies x=2 (mod 3), x= 1 (mod 4), and x= 3 (mod 5) using the method of back substitution
step 1: the first congruence can be written as x= 3t +2 where t is an integer. Substituting this expression for x into the second congruence gives 3t +2=1(mod4) step 2: This implies t=1(mod4). Therefore, t=4u+1 for some integer u. Step 3: thus, x=3t +2=3(4u+1) +2= 12u+5. We substitute this into the third congruence to obtain 12u+5=3(mod5), which implies u=4(mod5) Step 4: Hence, u=5v+4 and so x=12u+5=12(5v+4)+5=70v+53, where v is an integer. Step 5: translating x=60v+53 back into a congruence, we find the solution to the simultaneous congruences x=53 (mod 60).
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? (45, 60)
the equation is 15 = (-1) * 45 +60
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? (8,9)
the equations is 1=(-1)* 8 + 1* 9
There are 16 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