gcd and lcm - using extended euclidean algorithm to find s, t, r

i am stuck for many hours and i don't understand using the extended euclidean algorithm. i calculated it the gcd using the regular algorithm but i don't get how to calculate it properly to obtain s,t,r.

i understand that from the gcd i can get a linear combination representation, but i don't get how to do it using the algorithm.

how can i find $s,t,r$ for $a=154, b= 84$?

if it is of any importance, the algorithm i am referring to is from the book cryptography: theory and practice

thank you very much. became hopeless because of it

Answer

Using the Euclidean algorithm, we have

$$\begin{align} 154&=1\cdot84+70\tag{1}\\ 84&=1\cdot70+14\tag{2}\\ 70&=5\cdot14+0 \end{align}$$

The last nonzero remainder is 14. So $\gcd(154,84)=14$. Now
$$\begin{align*} 14&=84-70\qquad\text{(using 2)}\\ &=84-(154-84)\qquad\text{(using 1)}\\ &=2\cdot84-1\cdot154 \end{align*}$$
So $14=2\cdot84-1\cdot154$.