11 - (i The linear Diophantine Equation ax by = c has a...

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Theorem. Let a, b, c Z . If c | ab and gcd ( a, c ) = 1, then c | b . Proof. Since gcd ( a, c ) = 1, there exist x, y Z s.t. 1 = ax + cy. Then b = ( ab ) x + cby. Since c | ( ab ) and c | ( cb ), we have c | b. 1
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Linear Diophantine Equations of one variable: ax = b, where a, b Z are given, a = 0, x Z is a variable. of two variables: ax + by = c, where a, b, c Z are given, a = 0 , b = 0, x, y Z are variables. 2
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Linear Diophantine Equation Theorem.
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Unformatted text preview: (i) The linear Diophantine Equation ax + by = c has a solution if and only if gcd( a, b ) | c. (ii) If x = x and y = y is one particular solu-tion, then the complete solution is x = x + m b d , y = y-m a d for all m ∈ Z . 3...
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