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$\begingroup$

$$ 3f + 280y = 1 $$

euclidean algorithm

$$ 280 = 3\cdot 93 + 1 $$
$$ 3 = 1 \cdot 3 $$

Back sub

$$ 1 = 280 + 3(-93) $$

my question is why a negative sign for 93? since in the euclidean algorithm the 93 is a positive one.

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    Because in $280 = 3\cdot 93 + 1$ you moved $3\cdot 93$ on the other side: $280 - 3\cdot 93 = 1$.2011-05-02

2 Answers 2

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In the Euclidean Algorithm finding the solutions to $$3f+280y=1$$ is equivalent to finding the solutions to $$3a-280b=1$$ And it's not abnormal to have solutions that are negative since solutions to the gcd equation are not unique.

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There's a negative sign because $1=280+3(93)$ is false.