Find the greatest common divisor of $10293$ and $29384$.
To find the greatest common divisor (GCD) of two numbers, I need to find the largest positive integer that divides both numbers evenly. One way to do this is to use the Euclidean algorithm, which repeatedly applies the fact that the GCD of two numbers does not change if I subtract the smaller number from the larger one. So, I start by subtracting $10293$ from $29384$, which gives $19091$. Then, I subtract $10293$ from $19091$, which gives $8798$. Then, I subtract $8798$ from $10293$, which gives $1495$. Then, I subtract $1495$ from $8798$, which gives $7303$. Then, I subtract $1495$ from $7303$, which gives $5808$. Then, I subtract $1495$ from $5808$, which gives $4313$. Then, I subtract $1495$ from $4313$, which gives $2818$.
[*. Then, I subtract $1495$ from $2818$, which gives $1323$. Then, I subtract $1323$ from $1495$, which gives $172$. Then, I subtract $172$ from $1323$, which gives $1151$. Then, I subtract $172$ from $1151$, which gives $979$. Then, I subtract $172$ from $979$, which gives $807$. Then, I subtract $172$ from $807$, which gives $635$. Then, I subtract $172$ from $635$, which gives $463$. Then, I subtract $172$ from $463$, which gives $291$. Then, I subtract $172$ from $291$, which gives $119$. Then, I subtract $119$ from $172$, which gives $53$. Then, I subtract $53$ from $119$, which gives $66$. Then, I subtract $53$ from $66$, which gives $13$. Then, I subtract $13$ from $53$, which gives $40$. Then, I subtract $13$ from $40$, which gives $27$. Then, I subtract $13$ from $27$, which gives $14$. Then, I subtract $13$ from $14$, which gives $1$. Now, I have reached $1$, which is the smallest positive integer and the GCD of any two numbers. Therefore, the GCD of $10293$ and $29384$ is $1$. 