Suppose $(a,b)$ are arguments of the euclidian algorithm, and $a>b$. Let's call $a_i$, $b_i$ their values on the i-th step of computation (We shall put $a$, $b$ = $a_0$,$b_0$):
$$ \begin{matrix} a_0 = & q_0 \cdot b_0 + r_0 \\ a_1 = & q_1 \cdot b_1 + r_1 \\ a_2 = & q_2 \cdot b_2 + r_2 \\ \dots \end{matrix} $$
Then, given that $a$ have decreased by the factor 5 after two steps of euclidian algorithm, find the relation $\frac {a_2} {b_2}$ provided that $b_2$ is non-zero.
Playing around the task I was unable to come up with $(a,b)$ pair which would comply to the provided conditions, namely the $b_2$ always turned into zero. Then I tried to write it in as a set of equations: $$ \begin{bmatrix} 5x & = q_0\ \cdot\ b + x \\ b & = q_1\ \cdot\ x + r_1 \\ x & = q_2\ \cdot\ r_1 + 0 \end{bmatrix} $$ So for the answer it is supposedly sufficient to find $q_2$, yet I am stuck.
Edit
Actually, I solved it, taking arbitrary $q_2=3$, $r_1=2$ and working it up. Hence answer is $3$, but it seem to be not unique answer. If I will be able to reach author and get some context on a problem, I will comment later.