# lecture24 - MA1100 Lecture 24 Revision Lecture 1 Overview...

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1 MA1100 Lecture 24 Revision Lecture

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Lecture 24 2 Overview Proofs Mathematical Induction Logic Sets Functions Relations Number Theory Cardinalities
Lecture 24 3 Cardinality Relations and differences among infinite, denumerable, countable and uncountable sets. •Some standard countable and uncountable sets among the sets Z , Q , R and their subsets. Constructing a bijection from one given set to another. • Constructing countable/uncountable sets using set operations . What do you need to know?

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Lecture 24 4 Number Theory Divisibility Congruence Modulo GCD Prime numbers Euclidean Algorithm Linear combination Prime Factorization (FTA) Infinitude of primes Types of primes Relatively Prime Number/sum of Divisors Perfect squares Division Algorithm Perfect numbers Square-free numbers
Lecture 24 5 Divisibility Useful arguments 1. a | b b = ak for some integer k 2. a | b 3. a | b 4. a | b and a | c 5. a | b and a | c 6. a | b and b | c 7. a | b and b | a a | b c a | bn for all integers n ac | bc for all integers n a | c a = b a | gcd(b, c)

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Lecture 24 6 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 4. a ª b mod n 5. m | n and a ª b mod n 6. a ª b mod n and c ª d mod n 7. a ª b mod n and c ª d mod n 8. a ª b mod n 9. r is the remainder of a when divided by n, 10. a and b has same remainder when divided by n, 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
Lecture 24 7 GCD Useful arguments 1. d = gcd(a, b) 2. d = gcd(a, b) 3. d | a and d | b 4. ax + by = d for some integers x,y 5. gcd(a, 0) = 6. gcd(a, b) = 7. gcd(a, b) = d d | a and d | b ax + by = d for some integers x,y d | gcd(a, b) d | gcd(a, b) |a| gcd(a, a b) gcd(a/d, b/d) = 1

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Lecture 24 8 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 4. a | bc and gcd(a, b) = 1 5. a | c and b | c and gcd (a, b) = 1 gcd(a, b) = 1 ax + by = 1 for some integers x,y p does not divide a gcd(b, c) = 1 a | c ab | c
Lecture 24 9 Prime numbers 1. d | p 2. p | a 3. p | ab 4. p | a 1 a 2 …a n 5. There are infinitely many prime numbers 6. 2 n - 1 is a prime 7. 2 n + 1 is an odd prime 8. n > 1 is not a prime Let p be a prime. d = 1 or d = p gcd(a, p) = p p | a or p | b p | a k for some k n is a prime n is a power of 2 p | n for some prime p Given any n, there is a prime p > n Useful arguments

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10 Prime factorization FTA: Every composite number has a unique prime factorization Applications Find GCD Criteria for perfect square Criteria for certain irrational numbers Criteria for divisibility Find number and sum of divisors Find special divisors of integers If
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## This note was uploaded on 04/15/2010 for the course MATHS MA1101R taught by Professor Vt during the Spring '10 term at National University of Singapore.

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lecture24 - MA1100 Lecture 24 Revision Lecture 1 Overview...

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