Question in fraction (not simple )

Question

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

Answer 1

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

$$\frac{37}{13}=2+\dfrac{1}{1+\dfrac{1}{5+\dfrac{1}{2}}}$$

Answer 2

\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 $$ y = \frac{13 - 11}{-54 + 55} = 2. $$