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 e.g., R₂ = 11, R3 = 111, R4 = Rn = 1111, etc. Hint: 10 Rn = n - 1 1 10 Hint: The case where 3 does not divide a is easier.
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 e.g., R₂ = 11, R3 = 111, R4 = Rn = 1111, etc. Hint: 10 Rn = n - 1 1 10 Hint: The case where 3 does not divide a is easier.
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter1: Fundamental Concepts Of Algebra
Section1.2: Exponents And Radicals
Problem 92E
Related questions
Question
[Number Theory] How do you solve question 5? 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](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F95d67d68-d7ff-4103-8924-55da21d6141d%2Faf547c50-0cec-49e3-a807-68fffb9d4944%2Fvq2zezg_processed.png&w=3840&q=75)
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
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