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Precalculus Mathematics Homework Help
Direct Proof of gcd(a,b) Corollary: ax+by=d
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[QUOTE="Miike012, post: 4537103, member: 265497"] Corollary from book: if d= gcd(a,b), then there exists integers x and y such that ax + by = d. This is not an obvious statement to me. Are there any direct proofs to prove this statement? The book proves this by induction. My proof: Suppose d = gcd(a,b) and a and b are positive integers. a does not necessarily divide b and b does not necessarily divide a so let q[SUB]a[/SUB] and r[SUB]a[/SUB] be integers such that a = q[SUB]a[/SUB]b + r[SUB]a[/SUB]. If d|a then d|(q[SUB]a[/SUB]b + r[SUB]a[/SUB]) therefore d|r[SUB]a[/SUB], that is there exists an integer R[SUB]a[/SUB] such that r[SUB]a[/SUB] = R[SUB]a[/SUB]d. Let q[SUB]b[/SUB] and r[SUB]b[/SUB] be integers such that b = q[SUB]b[/SUB]a + r[SUB]b[/SUB]. We can see d|r[SUB]b[/SUB] so Let R[SUB]b[/SUB] be the integer such that r[SUB]b[/SUB] = R[SUB]b[/SUB]d. now a + b = q[SUB]a[/SUB]b + q[SUB]b[/SUB]a + (R[SUB]a[/SUB] + R[SUB]b[/SUB])d and a(1-q[SUB]b[/SUB])/(R[SUB]a[/SUB] + R[SUB]b[/SUB]) + b(1-q[SUB]a[/SUB])/(R[SUB]a[/SUB] + R[SUB]b[/SUB]) = d. Now I just need to prove that the coef. of a and b are integers... [/QUOTE]
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Direct Proof of gcd(a,b) Corollary: ax+by=d
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