Can you do #9? ler's PhiExercises for Section 4.1. 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?5. Use Fermat's Theorem to show that n13- n is divisible by 2730 for all n.6. Show that if p > 3 is prime, then ab"baP is divisible by 6p.-7. Show, using the Binomial Theorem, that if p is prime and a and b are inte-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 theform n2 + 1.9. Show that there are infinitely many primes of the form 16k + 1. Moregenerally, show that for any r > 0, there are infinitely many primes of theform 2" k + 1.10. Let n = r4 +1. Show that 3, 5, and 7 cannot divide n. What is the smallestprime 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 Mersennenumber 2P – 1 is of the form 1 + 2pk for some k.12. What can you say about the prime factors of a composite Fermat numberFn = 22" +1? Use Fermat's Theorem and Proposition 4.1.5 to find a factorof 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 integersolutions withpł xyz, thenp satisfies 2P! = 1 (mod p²). A prime psatisfying this latter congruence is called a Wiefrich prime. Determine thefirst two Wiefrich primes.

Name: Can you do #9? ler's PhiExercises for Section 4.1. Use Fermat's Theore
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Description: Can you do #9? ler's PhiExercises for Section 4.1. 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 d

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Answered: 9. Show that there are infinitely many…

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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