2. Show that 1184 – 534 is divisible by 7. -

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1.2 Euler's Phi Function
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.
Tb
2. Show that 1184 – 584 is divisible by 7.
3. Show that ifn = 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 n
13
n is divisible by 2730 for all
baP is divisible by 6p.
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. Mo
generally, show that for any r > 0, there are infinitely many primes of th
form 2" k + 1.
10. Let n = r4 +1. Show that 3, 5, and 7 cannot divide n. What is the smalle:
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 Mersenn
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 numbe
Fn = 22" +1? Use Fermat's Theorem and Proposition 4.1.5 to find a facto
of F5, thereby disproving Fermat's statement that all the Fn are prime.
13. In 1909, Wiefrich proved that if p is prime and x? + y :
solutions with p ł ruz, then n satisfier ?P-1
zP has integer
Transcribed Image Text:1.2 Euler's Phi Function 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. Tb 2. Show that 1184 – 584 is divisible by 7. 3. Show that ifn = 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 n 13 n is divisible by 2730 for all baP is divisible by 6p. 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. Mo generally, show that for any r > 0, there are infinitely many primes of th form 2" k + 1. 10. Let n = r4 +1. Show that 3, 5, and 7 cannot divide n. What is the smalle: 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 Mersenn 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 numbe Fn = 22" +1? Use Fermat's Theorem and Proposition 4.1.5 to find a facto of F5, thereby disproving Fermat's statement that all the Fn are prime. 13. In 1909, Wiefrich proved that if p is prime and x? + y : solutions with p ł ruz, then n satisfier ?P-1 zP has integer
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