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Propostion:
There do not exist integer $m$ and $n$ such that $12m+15n=1$

It seem obvious but I am not sure how to show this.

Proposition 2:
If there exist integer m and n such that $12M+15n=1$ then $m$ and $n$ are both positive.

Well clearly the antecent if false on this one. So nothing can really be proved.

But I am not sure how to show this.

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Proposition 2 is dealt with per ex falso quodlibet. For proposition 1, why does it seem obvious? –  Daniel Fischer Jun 16 at 18:03
    
$12m+15n = 3(4m+5n) \neq 1$, since there is no integer $z \in \Bbb Z$ such that $3z = 1$ –  Einer Jun 16 at 18:04

3 Answers 3

up vote 5 down vote accepted

$12m+15n=3(4m+5n)$ is always a multiple of $3$, but $1$ is not.

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No. By the Bezout's lemma $12m+15n=1\iff \gcd(12,15)=1$ which's obviously wrong.

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The left hand side is divisble by three, the right hand side is not.

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