Homework Help is Here – Start Your Trial Now!
Math
Advanced Math
1. Solve the congruence x² + 1 = 0 mod 13³.

1. Solve the congruence x² + 1 = 0 mod 13³.

Elements Of Modern Algebra
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 58E: a. Prove that 10n(1)n(mod11) for every positive integer n. b. Prove that a positive integer z is...
See similar textbooks
icon
Related questions
Question

[Number Theory] How do you solve question 1?  This means all integers x that satisfy the congruence. thanks

1. Solve the congruence 2. Solve the congruence x² + 1 = 0 mod 13³. x³ - 2x² - 7x+3 = 0 mod 7². 3. Suppose that n is an odd positive integer and write o(n) a b n with ged(a, b) = 1. Show that the largest prime factor of b is the largest prime factor of n 4. Find the smallest positive odd integer n such that o(n) 7680 n 12121 You should do this by a mathematical analysis not a brute force computer search. Hint: Use Problem 3. e.g., R₂ = 11, R3 = 111, R4 = 5. Suppose that a is a positive integer relatively prime to 10. Show that a divides infinitely many 'repunits', i.e., numbers of the form Rn = 1111, etc. Hint: Rn 11…….11. 10 - 1 10 1 Hint: The case where 3 does not divide a is easier. = n
Transcribed Image Text:1. Solve the congruence 2. Solve the congruence x² + 1 = 0 mod 13³. x³ - 2x² - 7x+3 = 0 mod 7². 3. Suppose that n is an odd positive integer and write o(n) a b n with ged(a, b) = 1. Show that the largest prime factor of b is the largest prime factor of n 4. Find the smallest positive odd integer n such that o(n) 7680 n 12121 You should do this by a mathematical analysis not a brute force computer search. Hint: Use Problem 3. e.g., R₂ = 11, R3 = 111, R4 = 5. Suppose that a is a positive integer relatively prime to 10. Show that a divides infinitely many 'repunits', i.e., numbers of the form Rn = 1111, etc. Hint: Rn 11…….11. 10 - 1 10 1 Hint: The case where 3 does not divide a is easier. = n
Expert Solution
steps

Step by step

Solved in 4 steps with 3 images

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer