(1) Show that the congruence .52 E a. (mod 2'9), where a. is odd, 6: is an integer, e Z 3, has either no solutions or exactly four incongruent solutions. (Hint: Use the fact that (:l:;3)2 E (291 i at)2 (mod 26).) (2) Show that if n. is a positive integer with n. 3 2, then an. does not divide 2'\" 1. (3) Show for any prime p, if a)\" E ()3) (mod p), then a? E b?" (mod 132). (4) Show there are infinitely many positive integers in. such that nl+1 is divisible by at least two distinct primes. (5) Explain why we should not choose primes p and q that are too close together to form the modulus n. in the RSA cryptosystem. In particular, show that using a pair of twin primes for p and q : p + 2 would be disastrous (Hint: recall Fermat's factorization method). The following questions are probably most easily done using a computer algebra system such as SageMath, or cloud systems such as CoCalc or WolframAlpha. (6) Find the primes p and q if n : pq : :1, 386,607 and 3(-n..) : 41,382,136. (7) lfthe ciphertext produced by RSA encryption with the key (9, n.) : (:3, 2881) is 0:304 1874 0347 051:3 2088 23:36 0736 0468, what is the plain text message? (8) Use the Pollard p 1 factorization method to find a nontrivial divisor of 8131