3. Show that ifn = 2 (mod 4), then 9" + 8" is divisible by 5.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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Contingency Table
A contingency table can be defined as the visual representation of the relationship between two or more categorical variables that can be evaluated and registered. It is a categorical version of the scatterplot, which is used to investigate the linear relationship between two variables. A contingency table is indeed a type of frequency distribution table that displays two variables at the same time.
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 #3?
![Exercises for Section 4.1
1. Use Fermat's Theorem to compute the following quantities.
(a) 31100 mod 19.
(b) 210000 mod 29.
(c) 99999 mod 31.
-2. Show that 1184 – 584 is divisible by 7.
3. Show that if n = 2 (mod 4), then 9" + 8" is divisible by 5.
4. For which values of n is 3" + 2" divisible by 13? by 7?
n is divisible by 2730 for all -
ba? is divisible by 6p.
5. Use Fermat's Theorem to show that n13
6. Show that if p> 3 is prime, then ab"
7. Show, using the Binomial Theorem, that if p is prime and a and b are int
gers,
then (a + b)P = a + b (mod p).
8. Show that no prime number of the form 4k + 3 can divide a number of th
form n2 + 1.
9. Show that there are infinitely many primes of the form 16k + 1. More
generally, show that for any r > 0, there are infinitely many primes of the
form 2" k + 1.
10. Let n = r4 +1. Show that 3, 5, and 7 cannot divide n. What is the smallest
prime that can divide n? Determine the form of the prime divisors of n.
11. Show that any proper factor, whether prime or not, of a composite Mersenne
number 2P – 1 is of the form 1 + 2pk for some k.
12. What can you say about the prime factors of a composite Fermat number
Fn = 22" +1? Use Fermat's Theorem and Proposition 4.1.5 to find a factor
of F5, thereby disproving Fermat's statement that all the Fn are prime.
13. In 1909, Wiefrich proved that if p is prime and xP + yP = zP has integer
solutions withpt ryz, then p satisfies 2P-1 = 1 (mod p²). A prime p
satisfying this latter congruence is called a Winfrinh
first two](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F18423e9a-daba-4c36-8402-a49704983186%2F40e52298-40e7-4c51-9a98-05aa50bbc6a5%2F53n3vea_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Exercises for Section 4.1
1. Use Fermat's Theorem to compute the following quantities.
(a) 31100 mod 19.
(b) 210000 mod 29.
(c) 99999 mod 31.
-2. Show that 1184 – 584 is divisible by 7.
3. Show that if n = 2 (mod 4), then 9" + 8" is divisible by 5.
4. For which values of n is 3" + 2" divisible by 13? by 7?
n is divisible by 2730 for all -
ba? is divisible by 6p.
5. Use Fermat's Theorem to show that n13
6. Show that if p> 3 is prime, then ab"
7. Show, using the Binomial Theorem, that if p is prime and a and b are int
gers,
then (a + b)P = a + b (mod p).
8. Show that no prime number of the form 4k + 3 can divide a number of th
form n2 + 1.
9. Show that there are infinitely many primes of the form 16k + 1. More
generally, show that for any r > 0, there are infinitely many primes of the
form 2" k + 1.
10. Let n = r4 +1. Show that 3, 5, and 7 cannot divide n. What is the smallest
prime that can divide n? Determine the form of the prime divisors of n.
11. Show that any proper factor, whether prime or not, of a composite Mersenne
number 2P – 1 is of the form 1 + 2pk for some k.
12. What can you say about the prime factors of a composite Fermat number
Fn = 22" +1? Use Fermat's Theorem and Proposition 4.1.5 to find a factor
of F5, thereby disproving Fermat's statement that all the Fn are prime.
13. In 1909, Wiefrich proved that if p is prime and xP + yP = zP has integer
solutions withpt ryz, then p satisfies 2P-1 = 1 (mod p²). A prime p
satisfying this latter congruence is called a Winfrinh
first two
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