2. (a) If gcd(a , 35) = 1, show that a2 = 1 (mod 35). [Hint: From Fermat's theorem a = 1 (mod 7) and at = 1 (mod 5).] (b) If gcd(a , 42) = 1, show that 168 = 3. 7.8 divides a – 1. (c) If gcd(a, 133) = gcd(b, 133) = 1, show that 133 |a18 - b18 6.
2. (a) If gcd(a , 35) = 1, show that a2 = 1 (mod 35). [Hint: From Fermat's theorem a = 1 (mod 7) and at = 1 (mod 5).] (b) If gcd(a , 42) = 1, show that 168 = 3. 7.8 divides a – 1. (c) If gcd(a, 133) = gcd(b, 133) = 1, show that 133 |a18 - b18 6.
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
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Number 2
![pbox.com/recents? tk=web_left_nav_bar&preview-David+M.+Burton+-+Elementary+number+the
F Workflows
...
1. Use Fermat's theorem to verify that 17 divides 11104 + 1.
2. (a) If gcd(a , 35) = 1, show that a12 = 1 (mod 35).
[Hint: From Fermat's theorem a6
(b) If gcd(a , 42) = 1, show that 168 = 3-7. 8 divides a - 1.
(c) If gcd(a, 133) = gcd(b, 133) = 1, show that 133 a18 -b18.
3. From Fermat's theorem deduce that, for any integer n > 0, 13| 1112n-
4. Derive each of the following congruences:
(a) a = a (mod 15) for all a.
[Hint: By Fermat's theorem, a = a (mod 5).]
(b) a' =a (mod 42) for all a.
(c) a =a (mod 3-7. 13) for all a.
(d) a' = a (mod 30) for all a.
5. If gcd(a, 30) = 1, show that 60 divides a + 59.
6. (a) Find the units digit of 3 00 by the use of Fermat's theorem.
= 1 (mod 7) and a = 1 (mod 5).]
12n+6](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F048faa6b-a100-4249-8bc6-0274c2c51311%2Fe7b1c393-9ebb-429a-9477-59c557f8d6a0%2Fje512d_processed.jpeg&w=3840&q=75)
Transcribed Image Text:pbox.com/recents? tk=web_left_nav_bar&preview-David+M.+Burton+-+Elementary+number+the
F Workflows
...
1. Use Fermat's theorem to verify that 17 divides 11104 + 1.
2. (a) If gcd(a , 35) = 1, show that a12 = 1 (mod 35).
[Hint: From Fermat's theorem a6
(b) If gcd(a , 42) = 1, show that 168 = 3-7. 8 divides a - 1.
(c) If gcd(a, 133) = gcd(b, 133) = 1, show that 133 a18 -b18.
3. From Fermat's theorem deduce that, for any integer n > 0, 13| 1112n-
4. Derive each of the following congruences:
(a) a = a (mod 15) for all a.
[Hint: By Fermat's theorem, a = a (mod 5).]
(b) a' =a (mod 42) for all a.
(c) a =a (mod 3-7. 13) for all a.
(d) a' = a (mod 30) for all a.
5. If gcd(a, 30) = 1, show that 60 divides a + 59.
6. (a) Find the units digit of 3 00 by the use of Fermat's theorem.
= 1 (mod 7) and a = 1 (mod 5).]
12n+6
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