(2) Show that if a, b = Z/nZ has multiplicative orders k and 1 respectively and...

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(2) Show that if a, b = Z/nZ has multiplicative orders k and 1 respectively and (k,l) = 1, then ab has multiplicative order kl. Hint: If (ab)" = 1 mod n then a" is the inverse of br modulo n. Use the previous exercise with m = -1 to deduce that the order of a is the same as the order of br modulo n, and thus that r must be a multiple of kl. (3) Find the order of every element in G17. (4) Find (365), (68), (945), (27). (5) Compute and justify your answer (guessing the pattern will not do): (a) The last digit of 6535. (b) 2123 mod 15. (c) 665 mod 13. (6) You are using the RSA system to receive information. You constructed your code using the primes p = = 103 and q = 109, so that n = pq = 11227 and you picked a = = 5. (a) Why is this choice of a suitable? (b) You receive the following message /8156/4465/2914/1838/2771/. Determine the original message. You may use a computer. (2) Show that if a, b = Z/nZ has multiplicative orders k and 1 respectively and (k,l) = 1, then ab has multiplicative order kl. Hint: If (ab)" = 1 mod n then a" is the inverse of br modulo n. Use the previous exercise with m = -1 to deduce that the order of a is the same as the order of br modulo n, and thus that r must be a multiple of kl. (3) Find the order of every element in G17. (4) Find (365), (68), (945), (27). (5) Compute and justify your answer (guessing the pattern will not do): (a) The last digit of 6535. (b) 2123 mod 15. (c) 665 mod 13. (6) You are using the RSA system to receive information. You constructed your code using the primes p = = 103 and q = 109, so that n = pq = 11227 and you picked a = = 5. (a) Why is this choice of a suitable? (b) You receive the following message /8156/4465/2914/1838/2771/. Determine the original message. You may use a computer.