Can you do #3? (a) 3340 mod 341(b) 78° mod 100(c) 210000 mod 1212. Implement the algorithm for exponentiation modulo m on a compaer Usethis to check the result of Exercise 4.1.1.3. Use Euler's Theorem (and the Chinese Remainder Theorem) to show thatn12 = 1 (mod 72) for all (n, 72) = 1.4. What is the smallest positive integer such that n^ = 1 (mod 100) for all(n, 100) = 1?5. Prove that m$(n) + n®(m) = 1 (mod mn) if (m, n) = 1.6. Show that ifn = pq, a product of distinct primes, then a(n)+1(mod n) for all a.= anoionot1. Show that7. Suppose that n = rs withr > 2, s > 2 and (r, s)$(r)¢(s)a$(n)=1 (mod n), that is, a2 = 1 (mod n).8. Suppose n is the product of two odd primes, p and q. Letnun oA(n) %3D (р - 1) (q —1)/(p — 1,9 - 1).Show that a^(n) =1 (mod n) for all integers a, satisfying (a, n) = 1.9. 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).10. Prove that a560 = 1 (mod 561) for all (a, 561) = 1.11. Suppose a" = 1 (mod m) and aº = 1 (mod m); show that a(ª,y) = 1(mod m).f12. Determine whether the following integers are prime powers, and if so, methe prime power decomposition.(a) 24137569(b) 500246412961ivib(c) 486695567at

Name: Can you do #3? (a) 3340 mod 341(b) 78° mod 100(c) 210000 mod 1212. Imp
Uploaded: 2024-03-20T07:06:41.000Z
Channel: Bartleby
Description: Can you do #3? (a) 3340 mod 341(b) 78° mod 100(c) 210000 mod 1212. Implement the algorithm for exponentiation modulo m on a compaer Usethis to check the result of Exercise 4.1.1.3. Use Euler's Theorem (and the Chinese Remainder Theorem) to show thatn