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Identify, using the extended euclid algorithm:

$gcd(553,26)$

also the numbers:

$u,v \in \mathbb{Z}~~~~~~~~553u+26v=gcd(553,26)$

My calculation gives me $1$, so there isn't a real gcd, can someone approve this?

What's with $u$ and $v$ now? I can't solve this equation with any numbers $\in \mathbb{Z}$?

Thanks.

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    $\begingroup$ What do you mean by "real gcd"? If your calculation gives $1$, then it means that the gcd is $1$. $\endgroup$ – Wojowu Nov 11 '15 at 10:39
  • $\begingroup$ Probably that's just a term I made up. Because even of prime numbers a common divisor is 1, always. So I thought 1 is not a "real gcd". Just ignore that. $\endgroup$ – Lars Nov 11 '15 at 10:41
  • $\begingroup$ for the latter u need to use the extended euclidian algorithm see here: en.wikipedia.org/wiki/Extended_Euclidean_algorithm It is kind of using the euclidian algorithm 'backwards'. $\endgroup$ – Kees Til Nov 11 '15 at 10:52
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$$553=26\times21+7$$ $$26=3\times7+5$$ $$7=1\times5+2$$ $$5=2\times2+1$$ $$2=2\times1$$ Hence $(553,26)=1$

Now $$1=5-2\times2\\=5-2\times(7-5)\\=3\times5-2\times7\\=3(26-3.7)-2.7\\=3.26-11.7\\=3.26-11(553-26.21)\\=234\times26-11\times553$$

So $u=-11$ and $v=234$

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