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Math135Lecture13StudentNotes

# Math135Lecture13StudentNotes - Math 135 Lecture 13 Linear...

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Math 135: Lecture 13: Linear Congruences Definition 13.1. A relation of the form ax c (mod m ) is called a linear congruence in the variable x . A solution to such a linear congruence is an integer x 0 so that ax 0 c (mod m ) Equivalencies Recalling our table of statements equivalent to a b (mod m ), ax 0 c (mod m ) iff there exists y 0 such that Thus ax c (mod m ) has a solution ⇐⇒ there exists an integer x 0 such that ax 0 c (mod m ) ⇐⇒ there exists an integer y 0 such that ax 0 + my 0 = c ⇐⇒ gcd( a, m ) | c (by LDET, Part 1) Solutions In the notation of our current setting Recall LDET 2: Let gcd( a, m ) = d 6 = 0. If x = x 0 and y = y 0 is one particular integer solution to the linear Diophantine equation ax + my = c , then the complete integer solution is But then, if x 0 Z is one solution to ax c (mod m ) the complete solution will be Equivalently, x x 0 , x 0 + m d , x 0 + 2 m d , · · · , x 0 + ( d - 1) m d (mod m ) Note that there are distinct solutions modulo m .

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Math135Lecture13StudentNotes - Math 135 Lecture 13 Linear...

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