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Let set $D = \{(ax+by)\mid x,y \text{ are integers and } ax+by>0\}$ . Prove that $D$ is not empty.

I'm trying to prove the extended Euclidean algorithm without using the algorithm.

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$x=a$, $y=b$, but please edit the question into the body, not just the title. –  Gerry Myerson Feb 6 '13 at 0:57
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What if $a=b=0$? –  Robert Israel Feb 6 '13 at 1:05

1 Answer 1

You are asking us to prove the following:

Let set $D = \{(ax+by)\mid x,y \text{ are integers and } ax+by>0\}$ . Prove that $D$ is not empty.

Firstly, note that if $a = b = 0$ then this does not hold. So let us now assume without loss of generality that $a \neq 0$. Then we have two cases: either $a$ is positive or $a$ is negative. If $a$ is positive, then $1\cdot a + 0 \cdot b > 0$ and is in $D$. If $a$ is negative that $ -1 \cdot a + 0 \cdot b > 0$ and is in $D$. Regardless, we have a proof.

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