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Unformatted text preview: 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 so that ax c (mod m ) Equivalencies Recalling our table of statements equivalent to a b (mod m ), ax c (mod m ) iff there exists y such that Thus ax c (mod m ) has a solution there exists an integer x such that ax c (mod m ) there exists an integer y such that ax + my = 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 and y = y is one particular integer solution to the linear Diophantine equation ax + my = c , then the complete integer solution is But then, if x Z is one solution to ax c (mod m ) the complete solution will be Equivalently, x x ,x + m d ,x + 2 m d , ,x + ( d- 1) m d (mod m ) Note that there are distinct solutions modulo m ....
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