5. If a-l is the inverse of a modulo m and b-1 the inverse of b modulo m, show that a--1 is the inverse of ab modulo m. ot be on odd
5. If a-l is the inverse of a modulo m and b-1 the inverse of b modulo m, show that a--1 is the inverse of ab modulo m. ot be on odd
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter2: The Integers
Section2.5: Congruence Of Integers
Problem 39E
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Binomial Distribution
Binomial is an algebraic expression of the sum or the difference of two terms. Before knowing about binomial distribution, we must know about the binomial theorem.
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Can you do #5
![Julo
we see that
10. Write a computer program that determines if an element a is invertible mod-
to the congruence 3x = 7 (mod 17) is x = 8 (mod 17). Then the
7. If m is composite, what is the value (m – 1)! mod m in the standard residue
6.3-1.17 = 1, so the inverse of 3 modulo 17 is 6. Hence the solution
119/7= 17. Applying the extended Euclidean Algorithm.
solutions to 21x = 49 (mod 119) are 8, 25, 42, 59, 76, 93, and 110
Exercises for Section 3.2
1. Determine the invertible elements modulo 15, 17, and 32.
2. Determine the inverse of 67 modulo 119.
3. Determine all solutions to the congruences 7x = 5 (mod 13) and 4r =6
(mod 18).
4. Solve the following linear congruences.
(a) 11x = 28 (mod 37).
(b) 42x = 90 (mod 156).
J 5. If a-l is the inverse of a modulo m and b-1 the inverse of b modulo m,
show that a- is the inverse of ab modulo m.
6. Let p be an odd prime.
(a) What are the possible values of x mod p so that x is its own inverse
modulo p?
(b) Consider the product 1 · 2 . . (p – 1) = (p – 1)! (mod p). Show
pairing the elements with their inverses that (p – 1)! = -1 (mod P.
This result is known as Wilson's Theorem. (Does this match wiui
your conjecture in Exercise 3.1.23?)
system? Conclude that (m – 1)! = -1 (mod m) implies that m is pin
8. Use Exercise 6 to answer the following questions:
(a) Determine 65! mod 67.
(b) If p is a prime number, what is (p – 2)! mod p?
9. Show that if p = 4k + 1 is prime, then [(2k)!]² :
= -1 (mod p).
ulo m, and if it is, the program computes the inverse.
11. Write a computer program to solve linear congruences modulo t.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F18423e9a-daba-4c36-8402-a49704983186%2Fdf710e4d-f2a6-43c2-9b3d-80477ef67a1d%2F84oj8zm_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Julo
we see that
10. Write a computer program that determines if an element a is invertible mod-
to the congruence 3x = 7 (mod 17) is x = 8 (mod 17). Then the
7. If m is composite, what is the value (m – 1)! mod m in the standard residue
6.3-1.17 = 1, so the inverse of 3 modulo 17 is 6. Hence the solution
119/7= 17. Applying the extended Euclidean Algorithm.
solutions to 21x = 49 (mod 119) are 8, 25, 42, 59, 76, 93, and 110
Exercises for Section 3.2
1. Determine the invertible elements modulo 15, 17, and 32.
2. Determine the inverse of 67 modulo 119.
3. Determine all solutions to the congruences 7x = 5 (mod 13) and 4r =6
(mod 18).
4. Solve the following linear congruences.
(a) 11x = 28 (mod 37).
(b) 42x = 90 (mod 156).
J 5. If a-l is the inverse of a modulo m and b-1 the inverse of b modulo m,
show that a- is the inverse of ab modulo m.
6. Let p be an odd prime.
(a) What are the possible values of x mod p so that x is its own inverse
modulo p?
(b) Consider the product 1 · 2 . . (p – 1) = (p – 1)! (mod p). Show
pairing the elements with their inverses that (p – 1)! = -1 (mod P.
This result is known as Wilson's Theorem. (Does this match wiui
your conjecture in Exercise 3.1.23?)
system? Conclude that (m – 1)! = -1 (mod m) implies that m is pin
8. Use Exercise 6 to answer the following questions:
(a) Determine 65! mod 67.
(b) If p is a prime number, what is (p – 2)! mod p?
9. Show that if p = 4k + 1 is prime, then [(2k)!]² :
= -1 (mod p).
ulo m, and if it is, the program computes the inverse.
11. Write a computer program to solve linear congruences modulo t.
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Elements Of Modern Algebra
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Gilbert, Linda, Jimmie
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Elements Of Modern Algebra
Algebra
ISBN:
9781285463230
Author:
Gilbert, Linda, Jimmie
Publisher:
Cengage Learning,