answer all parts with full solution

(a) Recall in chapter 1 we proved mmk+r for all natural numbers k. Use the Euclidean Algorithm to prove indeed gcd(m,km+1)=1. (b) Prove there is a solution to the Diophantine equation ax+by=c iff the congruence axc (mod b ) has a solution, and then apply theory studied in chapters 45 to conclude the Diophantine equation has a solution when b is prime and b{a. (c) Solve 71x=1(mod120) by converting to a Diophantine equation. Note that 120 is not prime, so theory from chapters 45 does not apply. (d) RSA: Read the discussion of RSA on top of page 54, consider N=253 and the encrypter En =7. With this protocol, a person wants to send the message of M=13, what would they actually send us (R) ? And in another occasion someone sends us the message 37 . What was their intended message M ? Please show steps of your computations, even though calculators might be used for basic operations. (a) Recall in chapter 1 we proved mmk+r for all natural numbers k. Use the Euclidean Algorithm to prove indeed gcd(m,km+1)=1. (b) Prove there is a solution to the Diophantine equation ax+by=c iff the congruence axc (mod b ) has a solution, and then apply theory studied in chapters 45 to conclude the Diophantine equation has a solution when b is prime and b{a. (c) Solve 71x=1(mod120) by converting to a Diophantine equation. Note that 120 is not prime, so theory from chapters 45 does not apply. (d) RSA: Read the discussion of RSA on top of page 54, consider N=253 and the encrypter En =7. With this protocol, a person wants to send the message of M=13, what would they actually send us (R) ? And in another occasion someone sends us the message 37 . What was their intended message M ? Please show steps of your computations, even though calculators might be used for basic operations