Greatest Commom Divisor In my lecture notes there is the following: 
$$\newcommand{\gcd}{\operatorname{gcd}}
a, b \in \mathbb{N}, \gcd(a, b)=d \implies \exists s, t \in \mathbb{Z} : sa+tb=d \implies tb \equiv d \pmod a$$ 
If $\gcd(a, b)=1$ then $tb \equiv 1 \pmod a \ $ $ \ \ \ [b]_a \in \mathbb{Z}_a^{\star}$ and $[b]_a^{-1}=[t]_a$ 
$$r_0=a, r_1=b \\ r_0=q_1 r_1+r_2 , 0 < r_2 <r_1 \\ r_1=q_2 r_2 + r_3, 0<r_3 < r_2 \\ \dots \\ r_{l-1}=q_{l-1}r_{l-1}+r_l, 0<r_l < r_{l-1} \\ r_{l-1}=q_l r_l+0$$ 
$\gcd(a, b)=r_l$ 
$$b, b-1, b-2, \dots , 0 \\ l \leq b$$ 
Because there isn't any explanation what $b, b-1, b-2, \dots , 0$ means, do you have any idea about what it could mean? What does this represent?

My guess is the following: your teacher wanted to show that $l < b$. Since $r_1 = b$ and $r_{i+1} \leq r_i - 1$, it follows that $r_{i} \leq b - i + 1$ (which is what that sequence seems to represent). Now since $r_l > 0$, we get that $l \leq b$.
