1. Find the prime factorization of each of these integers. a) 39 b) 81 2. What...

 

 

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1. Find the prime factorization of each of these integers. a) 39 b) 81 2. What is the greatest common divisors of these pairs of integers? a) 2° * 33 + 55, 25 * 3 + 52 b) 17, 17'7 c) 22 * 7,53 * 13 3. What is the least common multiple of these pairs of integers? a) 22 * 33 * 55, 25 * 33 * 52 b) 17, 17'7 c) 22 * 7, 53 + 13 4. Use the Euclidean algorithm to find: a) gcd(100, 101) b) gcd(123, 127) 5. Find all values x such thatr =1 (mod 5), r= 2 (mod 6), and r = 3 (mod 7). 6. Decrypt the following ciphertext message: VFZWRRIIQQRCHUIHLRJVIIUWD, which was encrypted first using a shift cipher, then using a tranposition cipher. The key of the shift cipher was k = 3, and the transposition cipher was based on the permutation o of the set 1, 2 with o(1) = 2 and a(2) = 1. 7. Suppose you intercept the message 18050527030501 that has been ecrypted using the RSA algorithm. You know that the public key used for encryption is (33, 3). Decrypt the message to determine what was said. Use the process shown in lecture to complete this problem. 8. A particular brand of shirt comes in 12 different colors, has a male and female version, and comes in three sizes for each sex. How many different types of this shirt are there? 9. How many bit strings are there of length eight? 10. A bowl contains 10 red balls and 10 blue balls. A woman selects balls at random without looking at them. a) Ilow many balls must she select to be sure of having at least 3 balls of the same color? b) How many balls must she select to be sure of having at least 3 blue balls? 11. 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. 12. A computer network consists of six computers. Each computer is directly connected to at least one of the other computers. Show that there are at least two computers in the network that are directly connected to the same number of other computers. 1. Find the prime factorization of each of these integers. a) 39 b) 81 2. What is the greatest common divisors of these pairs of integers? a) 2° * 33 + 55, 25 * 3 + 52 b) 17, 17'7 c) 22 * 7,53 * 13 3. What is the least common multiple of these pairs of integers? a) 22 * 33 * 55, 25 * 33 * 52 b) 17, 17'7 c) 22 * 7, 53 + 13 4. Use the Euclidean algorithm to find: a) gcd(100, 101) b) gcd(123, 127) 5. Find all values x such thatr =1 (mod 5), r= 2 (mod 6), and r = 3 (mod 7). 6. Decrypt the following ciphertext message: VFZWRRIIQQRCHUIHLRJVIIUWD, which was encrypted first using a shift cipher, then using a tranposition cipher. The key of the shift cipher was k = 3, and the transposition cipher was based on the permutation o of the set 1, 2 with o(1) = 2 and a(2) = 1. 7. Suppose you intercept the message 18050527030501 that has been ecrypted using the RSA algorithm. You know that the public key used for encryption is (33, 3). Decrypt the message to determine what was said. Use the process shown in lecture to complete this problem. 8. A particular brand of shirt comes in 12 different colors, has a male and female version, and comes in three sizes for each sex. How many different types of this shirt are there? 9. How many bit strings are there of length eight? 10. A bowl contains 10 red balls and 10 blue balls. A woman selects balls at random without looking at them. a) Ilow many balls must she select to be sure of having at least 3 balls of the same color? b) How many balls must she select to be sure of having at least 3 blue balls? 11. 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. 12. A computer network consists of six computers. Each computer is directly connected to at least one of the other computers. Show that there are at least two computers in the network that are directly connected to the same number of other computers.