Find the greatest common divisor of $5616$ and $11609$.
We use the Euclidean algorithm.  \begin{align*}
\text{gcd}\,(5616,11609)&=\text{gcd}\,(5616 ,11609- 2 \cdot 5616) \\
&=\text{gcd}\,(5616, 377)\\
&=\text{gcd}\,(5616-14 \cdot 377,377)\\
&=\text{gcd}\,(338,377)\\
&=\text{gcd}\,(338,377-338)\\
&=\text{gcd}\,(338,39)\\
&=\text{gcd}\,(338 - 8 \cdot 39,39)\\
&=\text{gcd}\,(26,39).
\end{align*}We can easily find that the greatest common divisor of $26$ and $39$ is $\boxed{13}$.
13
Level 3
Number Theory
Problem: Find the greatest common divisor of $5616$ and $11609$.
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>To find the greatest common divisor (GCD) of two numbers, we can use the Euclidean algorithm. This method involves repeated division and finding remainders. 

First, we divide the larger number, $11609$, by the smaller number, $5616$. We calculate the quotient and the remainder:

$11609 \div 5616 = 2$ (quotient) with a remainder of $11609 - 2 \times 5616 = 11609 - 11232 = 377$.

Next, we replace the larger number with the smaller number and the smaller number with the remainder. Now we need to find the GCD of $5616$ and $377$.

We repeat the process:

$5616 \div 377 = 14$ (quotient) with a remainder of $5616 - 14 \times 377 = 5616 - 5278 = 338$.

Now we find the GCD of $377$ and $338$:

$377 \div 338 = 1$ (quotient) with a remainder of $377 - 1 \times 338 = 377 - 338 = 39$.

Next, we find the GCD of $338$ and $39$:

$338 \div 39 = 8$ (quotient) with a remainder of $338 - 8 \times 39 = 338 - 312 = 26$.

Now we find the GCD of $39$ and $26$:

$39 \div 26 = 1$ (quotient) with a remainder of $39 - 1 \times 26 = 39 - 26 = 13$.

Next, we find the GCD of $26$ and $13$:

$26 \div 13 = 2$ (quotient) with a remainder of $26 - 2 \times 13 = 26 - 26 = 0$.

When the remainder reaches $0$, the last non-zero remainder is the GCD. Here, the last non-zero remainder is $13$. 

Thus, the greatest common divisor of $5616$ and $11609$ is $13$.<|END_OF_LATENT|> The final answer is $13$.