Can you do #6 roc(a) 3340 mod 341mu(b) 78° mod 100havr1)(c) 210000 mod 121but! 2. Implement the algorithm for exponentiation modulo m on a compuer Usethis to check the result of Exercise 41.1.noncom3. Use Euler's Theorem (and the Chinese Remainder Theorem) to sho thatn1? = 1 (mod 72) for all (n, 72) = 1.Remmia4. What is the smallest positive integer A such that n^ = 1 (mod 100) for all(n, 100) = 1?hasf (xthe c5. Prove that m (m) + n°(m) = 1 (mod mn) if (m, n) = 1.6. Show that ifn = pq, a product of distinct primes, then a$(n)+1(mod n) for all a.= aThepolyrT= 1. Show thatod7. Suppose that n = rs withr > 2, s > 2 and (r, s) :р.$(n)(r)6(s)a=1 (mod n), that is, a^´ = 1 (mod n).comp8. Suppose n is the product of two odd primes, p and q. LetPROCX(n) = (p – 1)(q – 1)/(p – 1, q – 1).If f iShow that a^(n) =1 (mod n) for all integers a, satisfying (a, n) = 1.ах +no solhave t9. Use a computer to find the composite numbers n < 2000 such that 2" =2 (mod n). Repeat the exercise to find composite n such that 3" = 3(mod n).NowhoseLet f(has no10. Prove that a560 = 1 (mod 561) for all (a, 561) = 1.11. Suppose a² = 1 (mod m) and aº = 1 (mod m); show that a(¤;y) = 1(mod m).for pol12. Determine whether the following integers are prime powers, and if so, findthe prime power decomposition.(a) 24137569that is,(b) 500246412961NoTherefo(c) 486695567at mostdivisibleSuppose

Name: Can you do #6 roc(a) 3340 mod 341mu(b) 78° mod 100havr1)(c) 210000 mod
Uploaded: 2024-03-20T07:06:41.000Z
Channel: Bartleby
Description: Can you do #6 roc(a) 3340 mod 341mu(b) 78° mod 100havr1)(c) 210000 mod 121but! 2. Implement the algorithm for exponentiation modulo m on a compuer Usethis to check the result of Exercise 41.1.noncom3. Use Euler's Theorem (and the Chinese Remainder Th