MAT 312/AMS 351 Fall 2010 Review for Midterm 2 § 1.7. Understand that if ( a, n ) = 1, then the equation ax ≡ b mod n has a unique solution (mod n ) and know how to Fnd it. That is the simplest case. Example 1.68 p.45. More generally, understand that if ( a, n ) = d , then the equation ax ≡ b mod n has d solutions (mod n ) if and only if d | b , and that all these solutions are congruent mod n/d . Know how to Fnd them. Example 1.69 p.45. Know how to apply the proof of the “Chinese Remainder Theorem” to solve a system of congruences. Example 1.71 p.46. Non-linear congruences will not be covered on the exam. § 1.8. Understand how to prove “±ermat’s Little Theorem” (p.51) and how to apply it. More generally, know the deFnition of ϕ ( n ) for any positive integer n (“the Euler ϕ-function”). Know how to calculate ϕ ( n ) given the prime factor decomposition of n (Theorem 1.86). Know how to apply Euler’s theorem: if ( a, n ) = 1, then a ϕ ( n ) ≡ 1 modulo n . § 1.9 Understand the principle behind the RSA method for public-key cryp-
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