Identify gcd(26,553) with extended euclid algorithm

Question

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.

Answer 1

$$553=26\times21+7$$ $$26=3\times7+5$$ $$7=1\times5+2$$ $$5=2\times2+1$$ $$2=2\times1$$ 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 $v=234$