MT1Summary - p divides ab and p is prime, either p divides...

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Course summary for Midterm I. The exam will cover sections 1.1 – 1.6 of the textbook dedicated to the number theory topics. The most important notions ans concepts are: Division. You should be able to divide integers, compute quotients and remainders, and use this knowledge in practice (for example, presenting the last digit of a number as its remainder modulo 10). The greatest common divisor. You should be able to find the greatest common divisor of two numbers and justify your answer using prime factorisation and Euclidean algorithm. You should be able to present it as a linear combination of these numbers using Euclidean algorithm or matrix method. Prime numbers and Unique Factorisation Theorem. You should be able to check if a given number is prime, present prime decompositions and use the properties of primes (for example, if
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Unformatted text preview: p divides ab and p is prime, either p divides a or p divides b ). Congruence classes. You should be able to perform arithmetic oper-ations with the congruence classes (addition, multiplication, powers), solve congruences ax = b mod n, (1) and systems of congruences (using Chinese Remainder Theorem) x = a mod m, x = b mod n. (2) The operations with powers of congruence classes can be made directly or with the usage of Fermats and Eulers theorems. Mathematical induction. You should be able to prove statements using induction, formulate the base case and the step of induction and prove them. Basics of public key codes. You may be asked to encode or decode a message using the Eulers Theorem and basic RSAtype algorithm. 1...
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This note was uploaded on 01/25/2012 for the course MAT 312 taught by Professor Bescher during the Summer '08 term at SUNY Stony Brook.

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