Is there a technique other than performing Euclid's algorithm in reverse that can elegantly show that if GCD$(a,b) = d$ then there exist integers $x$ and $y$ such that $ax + by = d$?
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HINT: Consider the smallest positive integer that can be written as $ax+by$
ronno
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1This is perfect. Thanks. But apparently there is a time limit on quickly an answer can be accepted so I up-voted for the moment. – dcdo Sep 30 '12 at 15:37
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@dcdo See here for full details and conceptual elaboration. – Bill Dubuque Sep 30 '12 at 16:05
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Sorry, I don't get it. Smallest positive integer can be a.1 + b.1 = a + b. Then? – sud_ Oct 12 '18 at 07:25
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@sud_ $x$ and $y$ don't have to be positive, just $ax+by$. If the smallest such number is $d'$, show that $d$ divides $d'$ and that $d'$ divides both $a$ and $b$. See the link in Bill's comment for more details. – ronno Oct 13 '18 at 12:12