The topics listed below are covered by Problem Sets 4–6;  in-class material, handouts, and on-line notes and notebooks; and the following sections of the text::

Sections 6.1–6.8 and 15.6

• “divides” ( | ) relation: definition, properties
• prime numbers: definition, testing for primality
• proof that there are infinitely many prime numbers
• Fundamental Theorem of Arithmetic and the prime factorization
• “Division Algorithm” (given n and d, there exist q, r with n = q d + r, 0 <= r < d); n mod d and n div d
• gcd: definition; recursive definition; finding from prime factorization; finding by Euclidean algorithm; expressing gcd(a, b) in form s a + t b
• relatively prime integers
• statement and proof  of more advanced properties divisibility, e.g., if a | bc and a is relatively prime to b, then a | c.
• congruence modulo m: definition; properties; Congruence Cancellation Law; modular arithmetic; the rings Z_m; finding multiplicative inverses modulo m by using the Euclidean algorithm
• application—affine ciphers: encoding letters & decoding numbers; choice of multiplicative key; cryptanalysis of additive, multiplicative, and affine ciphers by using frequency analysis
• Fermat’s Little Theorem and “Euler’s Corollary”
• application—public key cryptology: idea of public key systems; the RSA system (generation of keys; why it works; using the system to encrypt and decrypt)

You will have the use of Mathematica, but not of any prepared Mathematica notebooks.