Find the greatest common divisor of $40304$ and $30203$.
By the Euclidean Algorithm: \begin{align*}
\text{gcd}(40304, 30203) &= \text{gcd}(40304-30203, 30203) \\
&= \text{gcd}(10101, 30203) \\
&= \text{gcd}(30203-2\cdot10101, 10101) \\
&= \text{gcd}(10001, 10101) \\
&= \text{gcd}(10101 - 10001, 10001) \\
&= \text{gcd}(100, 10001) \\
&= \text{gcd}(10001 - 100\cdot100, 100) \\
&= \text{gcd}(1, 100) \\
\end{align*}Therefore, the greatest common divisor of $40304$ and $30203$ is $\boxed{1}$
1
Level 3
Number Theory
Problem: Find the greatest common divisor of $40304$ and $30203$.
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 taking remainders. 

First, we divide the larger number, $40304$, by the smaller number, $30203$. The quotient is $1$ and the remainder is calculated as follows: 

$40304 - 30203 \times 1 = 40304 - 30203 = 10101$.

Next, we apply the algorithm again, now using $30203$ and the remainder $10101$. We divide $30203$ by $10101$. The quotient is $2$ and the remainder is:

$30203 - 10101 \times 2 = 30203 - 20202 = 10001$.

We continue the process with $10101$ and $10001$. Dividing $10101$ by $10001$, we find the quotient is $1$ and the remainder is:

$10101 - 10001 \times 1 = 10101 - 10001 = 100$.

Next, we take $10001$ and $100$. Dividing $10001$ by $100$, the quotient is $100$ and the remainder is:

$10001 - 100 \times 100 = 10001 - 10000 = 1$.

Finally, we apply the algorithm one last time with $100$ and $1$. Dividing $100$ by $1$, the quotient is $100$ and the remainder is:

$100 - 1 \times 100 = 100 - 100 = 0$.

When the remainder reaches $0$, the last non-zero remainder is the GCD. In this case, the last non-zero remainder is $1$. 

Thus, the greatest common divisor of $40304$ and $30203$ is $1$.<|END_OF_LATENT|> The final answer is $1$.