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Math 470 Exam 1 Sample Problems September 22, 2011 Please note: These problems by no means make an exhaustive list of the types of problems that we have seen in class or on the homework or that you can expect to see on the exam. 1. Use the Euclidean Algorithm to find gcd(1333, 1591) and to find one solution of 1333x + 1591y = gcd(1333, 1591) with x, y ∈ Z. 2. Find all solutions to the equation 3x − 10y = 5 with x, y ∈ Z. 3. Find all congruence classes of solutions of the following congruences in the given modulus. (a) 7x ≡ 20 (mod 62). (b) 6x ≡ 3 (mod 32). (c) 4x ≡ 6 (mod 10). 4. What are the multiplicative inverses of 3, 5, 7, 9, and 15, modulo 26. 1 −2 5. Invert the matrix modulo 17. 9 7 6. We use the following alphabet: A 00 P 15 B 01 Q 16 C 02 R 17 D 03 S 18 E 04 T 19 F 05 U 20 G 06 V 21 H 07 W 22 I 08 X 23 J 09 Y 24 K 10 Z 25 L 11 . 26 M 12 , 27 N 13 ; 28 O 14 (a) Using the affine cipher x 7→ 15x + 22 (mod 29), encrypt the plaintext message HELP into its ciphertext. (b) Someone has sent you the message ELF using the encryption key in part (a). What did they say? 7. (a) Show that 2 is a primitive root modulo 11. (b) Find a primitive root of 17. 8. What are φ(16), φ(20), φ(31), and φ(36), where φ(n) is Euler’s φ-function? 9. Given that 881 is prime, use Fermat’s Little Theorem to simplify 101882 (mod 881). 10. You receive the following ciphertext. You know it was encoded using a Vigenè€re cipher, and you know that the key is a recognizable word or words. PIRHEETGSTPIRHEETGSTPIRHEETGSTPIRHEETGSTPIRHEETGST (a) What is a likely key length for this cipher? Explain. (b) You learn that the plaintext message consists of a two-letter word, one of whose letters is ‘a’, repeated over and over. What is the message and what is the key? (Do you want to revise your answer in (a)?)