Find all solutions: 561x + 909y = 81

Not just looking for answers, I don't understand the procedure. Any help is appreciated.

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- Feb 18th 2010, 04:53 PMZennie[SOLVED] Problem from a section on Bezout's Identity
Find all solutions: 561x + 909y = 81

Not just looking for answers, I don't understand the procedure. Any help is appreciated. - Feb 18th 2010, 06:27 PMtonio

Since $\displaystyle (561,909)=3$ and $\displaystyle 3\mid 81$ , there exists a solution in integers to the given equation :

Lemma: If $\displaystyle (m,n)=d \,\,\,and\,\,\,d\mid b\,,\,\,say\,\,\,b=kd$ then, after writing $\displaystyle d=ma+nb\,,\,\,a,b\in\mathbb{Z}$ ,we get that

$\displaystyle m(ka)+n(kb)=kd=b$ , and $\displaystyle mx+ny=b$ has an integer solution.

This lemma is in fact an iff lemma.

Tonio