How to express the rational numbers as finite continued fractions ?
with last term >1
1. 118/303
2. 187/57
3. 10001/10101
4. 12/240005
$\displaystyle
\frac{{187}}
{{57}} = 3 + \frac{{16}}
{{57}} = 3 + \frac{1}
{{{\raise0.5ex\hbox{$\scriptstyle {57}$}
\kern-0.1em/\kern-0.15em
\lower0.25ex\hbox{$\scriptstyle {16}$}}}} = 3 + \frac{1}
{{3 + {\raise0.5ex\hbox{$\scriptstyle 9$}
\kern-0.1em/\kern-0.15em
\lower0.25ex\hbox{$\scriptstyle {16}$}}}} = 3 + \frac{1}
{{3 + \tfrac{1}
{{1 + \tfrac{7}
{9}}}}}
$
$\displaystyle
\frac{{187}}
{{57}} = 3 + \frac{1}
{{3 + \tfrac{1}
{{1 + \tfrac{7}
{9}}}}} = 3 + \frac{1}
{{3 + \tfrac{1}
{{1 + \tfrac{1}
{{1 + \tfrac{2}
{7}}}}}}} = 3 + \frac{1}
{{3 + \tfrac{1}
{{1 + \tfrac{1}
{{1 + \tfrac{1}
{{3 + \tfrac{1}
{2}}}}}}}}}
$
Do you see how I did it?
Here is another approach you can take.
Use the Euclidean algorithm on $\displaystyle 187,57$:
$\displaystyle 187 = 3\cdot 57 + 16$
$\displaystyle 57 = 3\cdot 16 + 9$
$\displaystyle 16 = 1\cdot 9 + 7$
$\displaystyle 9 = 1\cdot 7+2$
$\displaystyle 7 = 3\cdot 2 + 1$
$\displaystyle 2 = 1\cdot 1+0$
Thus, $\displaystyle 187/57 = [3;3,1,1,3,1]$.
Now why does this not match what PaulRS did?
Probably a computational mistake somewhere, I do not have time to check it now.