Find the greatest common divisor of 9118, 12173, and 33182.
After identifying a factor of 2 in 9118 and 33,182, we find that the given integers appear to be difficult to prime factorize. Therefore, we turn to the Euclidean algorithm. To use the Euclidean algorithm to find the greatest common divisor of a set of three numbers, we first note that $\text{gcd}(a,b,c)=\text{gcd}(\text{gcd}(a,b),c)$ for integers $a$, $b$, and $c$. One way to see this is to consider the prime factorization of $a$, $b$, and $c$. Now we apply the Euclidean algorithm to the first pair of numbers to find \begin{align*}
\text{gcd}(9118,12,\!173) &= \text{gcd}(9118,12,\!173-9118) \\
&= \text{gcd}(9118,3055). \\
\end{align*}We could continue to apply the Euclidean algorithm as usual, but instead we note that $\text{gcd}(9118,3055)=\text{gcd}(9118,3055\div 5)$, since 9118 is not divisible by 5. We find \begin{align*}
\text{gcd}(9118,12,\!173) &= \text{gcd}(9118,3055 \div 5) \\
&= \text{gcd}(9118,611) \\
&= \text{gcd}(611,9118-611\times 15) \\
&= \text{gcd}(611,-47) \\
&= 47,
\end{align*}since long division shows that 611 is divisible by 47. Note that we chose 15 as the number of 611's to subtract from 9118 by dividing 9118 by 611 and rounding up to the nearest integer. Finally, we verify that 33,182 is divisible by 47. Altogether, we have \begin{align*}
\text{gcd}(9118, 12,\!173, 33,\!182) &= \text{gcd}(\text{gcd}(9118, 12,\!173), 33,\!182) \\ &= \text{gcd}(47, 33,\!182)\\ &=\boxed{47}.
\end{align*}
47
Level 4
Number Theory
Problem: Find the greatest common divisor of 9118, 12173, and 33182.
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>To find the greatest common divisor (GCD) of the numbers 9118, 12173, and 33182, we can use the Euclidean algorithm. This method involves repeated division and finding remainders.

First, we calculate the GCD of the first two numbers, 9118 and 12173. We divide 12173 by 9118:

12173 ÷ 9118 = 1 with a remainder of 3055.

Next, we apply the Euclidean algorithm again using 9118 and the remainder 3055:

9118 ÷ 3055 = 2 with a remainder of 3008.

We continue with 3055 and 3008:

3055 ÷ 3008 = 1 with a remainder of 47.

Next, we apply the algorithm to 3008 and 47:

3008 ÷ 47 = 63 with a remainder of 47.

Since the remainder is now 0, we conclude that the GCD of 9118 and 12173 is 47.

Now, we need to find the GCD of this result (47) and the third number, 33182. We check if 33182 is divisible by 47:

33182 ÷ 47 = 706 with a remainder of 0.

Since the remainder is 0, 47 divides 33182 evenly, confirming that the GCD of all three numbers is indeed 47.

Thus, the final answer is 47.<|END_OF_LATENT|> The final answer is $47$.