Lecture_20 - MA1100 Lecture 20 Number Theory Relatively...

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1 MA1100 Lecture 20 Number Theory Relatively Prime Prime numbers Prime Factorization
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MA1100 Lecture 20 2 Relatively Prime Definition Let a and b be two nonzero integers. If gcd(a, b) = 1 , we say that a and b are relatively prime . Remark Alternative term: co-prime In other words, a and b have no common divisors > 1 . The integers a and b need not be prime numbers in order to be relatively prime .
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MA1100 Lecture 20 3 Relatively Prime Example 1. 10 and 21 are relatively prime 2. Any two distinct primes are relatively prime 3.For any integer n, n and n+1 are relatively primes
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MA1100 Lecture 20 4 ax + by and Relatively Prime Proposition Let a and b be nonzero integers. a and b are relatively prime if and only if 1 is of the form ax + by for x, y œ Z . Proof a and b relatively prime ( Ø ) 1 is of the form ax + by for x, y œ Z . a and b not relatively prime ( ) 1 cannot be of the form ax + by for x, y œ Z .
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MA1100 Lecture 20 5 ax + by and Relatively Prime Example a = 11 and b = 17 are relatively prime . We may find a solution by inspection or reversing Euclidean Algorithm 11( -3 ) + 17( 2 ) = 1 For any n œ Z , 11x + 17y = n has integer solutions as every integer n is a multiple of 1 So 1 is of the form 11x + 17y for x, y œ Z . i.e. 11x + 17y = 1 has integer solutions in x and y . (x, y are variables here)
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MA1100 Lecture 20 6 ax + by and Relatively Prime For any integer a, gcd(2a + 1, 9a + 4) = 1 . Consider the equation (2a + 1)x + (9a + 4)y = 1 Proof (x, y are variables here) Proposition Show that this equation has integer solutions.
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MA1100 Lecture 20 7 Divisibility and Relatively Prime If c | a and gcd (a, b) = 1, then gcd(b, c) = 1 . Proposition Let a, b, c be any integers. So gcd(b, c) = 1. c | a gcd (a, b) = 1 Proof Since 1 can be written in the form of cn + bm,
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MA1100 Lecture 20 8 Divisibility and Relatively Prime True or False Let a, b, c be any integers. If a | bc , then a | b or a | c .
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Lecture 20 9 Divisibility and Relatively Prime Proposition Let a, b, c be any integers. If
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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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Lecture_20 - MA1100 Lecture 20 Number Theory Relatively...

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