Question in fraction (not simple ) I have a question and its answer but I don't know how can i solve 
$$\frac {37}{13} = 2+ \frac {1}{x+\frac{1}{5+\frac{1}{y}}} $$
the answer $ x =1, y=2$
Could any one explain how to solve this ?? please 
 A: This is a development in continued fraction.
At each step, you have to compute a quotient and remainder, like this:
$\frac{37}{13}$ has quotient $2$ and remainder $11$, thus $37=13 \times 2 + 11$, or also$\frac{37}{13}=2+\frac{11}{13}=2+\dfrac{1}{\dfrac{13}{11}}$.
Now you do the same with $\frac{13}{11}=1+\frac{2}{11}$, thus
$$\frac{37}{13}=2+\dfrac{1}{1+\dfrac{2}{11}}=2+\dfrac{1}{1+\dfrac{1}{\dfrac{11}{2}}}$$
And finally, $\dfrac{11}{2}=5+\dfrac{1}{2}$, thus
$$\frac{37}{13}=2+\dfrac{1}{1+\dfrac{1}{5+\dfrac{1}{2}}}$$
A: \begin{align*}
\frac {37}{13}
&= 2+ \frac {1}{x+\frac{1}{5+\frac{1}{y}}} \\
\frac {11}{13}
&= \frac {1}{x+\frac{1}{5+\frac{1}{y}}} \\
\frac {13}{11}
&= x+\frac{1}{5+\frac{1}{y}} \\
\frac {13 - 11x}{11}
&= \frac{1}{5+\frac{1}{y}} \\
\frac {11}{13 - 11x}
&= 5+\frac{1}{y} \\
\frac {-54 + 55x}{13 - 11x}
&= \frac{1}{y} \\
\frac {13 - 11x}{-54 + 55x}
&= y
\end{align*}
Assuming $x$ and $y$ must be integers, note that
$$
\gcd(13 - 11x, -54 + 55x) = \gcd(13-11x, 11) = \gcd(13, 11) = 1
$$
Hence
$$
-54 + 55x = \pm 1 \implies x = 1
$$
Then
$$
y = \frac{13 - 11}{-54 + 55} = 2.
$$
