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Unformatted text preview: 1 MA1100 Lecture 22 Revision Lecture • Number Theory • Functions • Relations MA1100 Lecture 22 2 Announcement ¡ Hand in homework 5 today ¢ Write your name, matric number and tutorial group on the cover page of your homework scripts. ¢ Marked HW scripts will be returned next week. ¡ Problem set 10 (available today) MA1100 Lecture 22 3 Countdown to Exam: 21 days Tut 10 Misc Tut 9 Number Th. Tutorial HW5 23. Exam Information 22. Revision lecture Week 12 Week 13 HW Fri Lecture Tue Lecture EXAM (NOV 25) Week 15 READING WEEK Week 14 MA1100 Lecture 22 4 After preparing your helpsheets, Ask yourself: “ Does the process of preparing the helpsheets help me gain better understanding of the topics, and see connections between concepts? ” MA1100 Lecture 22 5 Number Theory Concepts • Divisibility • Congruence Modulo • Greatest Common Divisor • Euclidean Algorithm • Integers of the form ax + by • Relatively Prime • Prime numbers • Prime Factorization (FTA) • Number of Divisors • Infinitude of primes • Diophantine equations MA1100 Lecture 22 6 Number Theory Divisibility Congruence Modulo GCD Prime numbers Diophantine equations Euclidean Algorithm Integers of the form ax + by Relatively Prime Prime Factorization Number of Divisors Infinitude of primes Remainder MA1100 Lecture 22 7 Divisibility Useful arguments 1. a  b ‹ b = ak for some integer k 2. a  b and a  c ﬂ 3. a  b ﬂ 4. a  b ﬂ 5. a  b and b  c ﬂ 6. a  b and b  a ﬂ a  b ≤ c a  bn for all integers n ac  bc for all integers c a  c a = ≤ b MA1100 Lecture 22 8 Congruence Modulo Useful arguments 1. a ª b mod n ‹ b = a + nk for some integer k 2. a ª 0 mod n ‹ 3. a ª b mod n and b ª c mod n ﬂ 4. a ª b mod n ﬂ 5. m  n and a ª b mod n ﬂ 6. a ª b mod n and c ª d mod n ﬂ 7. a ª b mod n and c ª d mod n ﬂ 8. a ª b mod n ﬂ 9. r is the remainder of a when divided by n, ﬂ 10. a and b has same remainder when divided by n, ñ n  a a ª c mod n ka ª kb mod kn for all positive k a ª b mod m a+c ª b+d mod n ac ª bd mod n a k ª b k mod n for all positive k a ª r mod n a ª b mod n MA1100 Lecture 22 9 GCD Useful arguments 1. d = gcd(a, b) ﬂ 2. d = gcd(a, b) ﬂ 3. d  a and d  b ﬂ 4. ax + by = d for some integers x,y ﬂ 5. gcd(a, 0) = 6. gcd(a, b) = 7. gcd(a, b) = d ﬂ d  a and d  b ax + by = d for some integers x,y d  gcd(a, b) gcd(a, b)  d a gcd(a, a ≤ b) gcd(a/d, b/d) = 1 MA1100 Lecture 22 10 Relatively prime Useful arguments 1. gcd(a, b) = 1 ñ 2. Given p is a prime. Then gcd(a, p) = 1 ñ 3. c  a and gcd(a, b) = 1 ﬂ 4. a  bc and gcd(a, b) = 1 ﬂ 5. a  c and b  c and...
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This note was uploaded on 03/19/2012 for the course SCIENCE MA1100 taught by Professor Forgot during the Fall '08 term at National University of Singapore.
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