| 模型 | 模型答案 | 正确答案 |
|---|---|---|
| 金声玉亮2.0 | To find the greatest common divisor (GCD) of the set \(\{322, -814, \frac{6821}{5}\}\), we need to follow these steps: 1. **Convert all numbers to integers**: Since the GCD is defined for integers, we need to convert \(\frac{6821}{5}\) to an integer. We can do this by multiplying all numbers by 5 to clear the fraction. \[ 322 \times 5 = 1610 \] \[ -814 \times 5 = -4070 \] \[ \frac{6821}{5} \times 5 = 6821 \] 2. **Find the GCD of the resulting integers**: Now we need to find the GCD of \(\{1610, -4070, 6821\}\). Since the GCD is always non-negative, we can ignore the negative sign and find the GCD of \(\{1610, 4070, 6821\}\). 3. **Use the Euclidean algorithm**: The Euclidean algorithm is a method for finding the GCD of two numbers. We will apply it step by step. - First, find the GCD of 1610 and 4070: \[ 4070 \div 1610 = 2 \quad \text{remainder} \quad 4070 - 2 \times 1610 = 850 \] \[ \text{GCD}(1610, 4070) = \text{GCD}(1610, 850) \] \[ 1610 \div 850 = 1 \quad \text{remainder} \quad 1610 - 850 = 760 \] \[ \text{GCD}(1610, 850) = \text{GCD}(850, 760) \] \[ 850 \div 760 = 1 \quad \text{remainder} \quad 850 - 760 = 90 \] \[ \text{GCD}(850, 760) = \text{GCD}(760, 90) \] \[ 760 \div 90 = 8 \quad \text{remainder} \quad 760 - 8 \times 90 = 40 \] \[ \text{GCD}(760, 90) = \text{GCD}(90, 40) \] \[ 90 \div 40 = 2 \quad \text{remainder} \quad 90 - 2 \times 40 = 10 \] \[ \text{GCD}(90, 40) = \text{GCD}(40, 10) \] \[ 40 \div 10 = 4 \quad \text{remainder} \quad 40 - 4 \times 10 = 0 \] \[ \text{GCD}(40, 10) = 10 \] - Now, find the GCD of 10 and 6821: \[ 6821 \div 10 = 682 \quad \text{remainder} \quad 6821 - 682 \times 10 = 1 \] \[ \text{GCD}(10, 6821) = \text{GCD}(10, 1) = 1 \] Therefore, the greatest common divisor of \(\{322, -814, \frac{6821}{5}\}\) is \(\boxed{1}\). |
\frac{1}{5} |
| deepseek-r1 | To find the greatest common divisor (GCD) of the set \(\left\{322, -814, \frac{6821}{5}\right\}\), we need to consider the GCD of the numerators divided by the least common multiple (LCM) of the denominators. First, we convert the integers to fractions with denominator 1: - \(322 = \frac{322}{1}\) - \(-814 = \frac{-814}{1}\) - \(\frac{6821}{5}\) is already a fraction. The numerators are \(322\), \(-814\), and \(6821\), and the denominators are \(1\), \(1\), and \(5\). 1. **Compute the GCD of the numerators \(322\), \(-814\), and \(6821\):** - Using the Euclidean algorithm for \(322\) and \(814\): - \(814 \div 322 = 2\) with remainder \(170\) - \(322 \div 170 = 1\) with remainder \(152\) - \(170 \div 152 = 1\) with remainder \(18\) - \(152 \div 18 = 8\) with remainder \(8\) - \(18 \div 8 = 2\) with remainder \(2\) - \(8 \div 2 = 4\) with remainder \(0\) - GCD of \(322\) and \(814\) is \(2\). - Now, compute GCD of \(2\) and \(6821\): - \(6821\) is odd, so GCD is \(1\). 2. **Compute the LCM of the denominators \(1\), \(1\), and \(5\):** - LCM of \(1\), \(1\), and \(5\) is \(5\). 3. **Combine the results:** - GCD of the numerators is \(1\). - LCM of the denominators is \(5\). - Therefore, the GCD of the set is \(\frac{1}{5}\). Thus, the greatest common divisor of the set \(\left\{322, -814, \frac{6821}{5}\right\}\) is \(\boxed{\dfrac{1}{5}}\). |
\frac{1}{5} |
| gpto1 | To find the greatest common divisor (GCD) of the set \(\{322, -814, \frac{6821}{5}\}\), we'll proceed step by step: **Step 1: Express all numbers with a common denominator.** First, write all numbers as fractions with denominator 5 (since \(\frac{6821}{5}\) has denominator 5). - \(322 = \frac{1610}{5}\) - \(-814 = \frac{-4070}{5}\) - \(\frac{6821}{5}\) remains the same. Now our set is \(\left\{\frac{1610}{5}, \frac{-4070}{5}, \frac{6821}{5}\right\}\). **Step 2: Find the GCD of the numerators.** Ignore the denominator for now and find the GCD of the numerators: 1610, 4070, and 6821. Using the Euclidean algorithm: 1. **Find GCD of 1610 and 4070:** \[ \begin{align*} 4070 &= 2 \times 1610 + 850 \\ 1610 &= 1 \times 850 + 760 \\ 850 &= 1 \times 760 + 90 \\ 760 &= 8 \times 90 + 40 \\ 90 &= 2 \times 40 + 10 \\ 40 &= 4 \times 10 + 0 \end{align*} \] So, \(\gcd(1610, 4070) = 10\). 2. **Find GCD of 10 and 6821:** \[ 6821 = 682 \times 10 + 1 \] So, \(\gcd(10, 6821) = \gcd(10, 1) = 1\). Therefore, the GCD of the numerators is **1**. **Step 3: Account for the common denominator.** Since the denominators are all 5, and the GCD of the numerators is 1, the GCD of the original numbers is \(\frac{1}{5}\). \[\boxed{\dfrac{1}{5}}\] |
\frac{1}{5} |
| gpt-4o-2024-08-06 | To find the greatest common divisor (GCD) of the set \(\{322, -814, \frac{6821}{5}\}\), we first need to ensure all numbers are integers. The number \(\frac{6821}{5}\) is not an integer, so we will convert it to an integer by multiplying all numbers by 5, the denominator of the fraction. First, calculate the integer equivalent of \(\frac{6821}{5}\): \[ \frac{6821}{5} \times 5 = 6821 \] Now, multiply the other numbers by 5: \[ 322 \times 5 = 1610 \] \[ -814 \times 5 = -4070 \] Thus, we need to find the GCD of the integers \(\{1610, -4070, 6821\}\). ### Step 1: Find the GCD of 1610 and 4070 First, find the GCD of 1610 and 4070 using the Euclidean algorithm: \[ 4070 \div 1610 \approx 2.528 \quad \Rightarrow \quad 4070 = 1610 \times 2 + 850 \] \[ 1610 \div 850 \approx 1.894 \quad \Rightarrow \quad 1610 = 850 \times 1 + 760 \] \[ 850 \div 760 \approx 1.118 \quad \Rightarrow \quad 850 = 760 \times 1 + 90 \] \[ 760 \div 90 \approx 8.444 \quad \Rightarrow \quad 760 = 90 \times 8 + 40 \] \[ 90 \div 40 \approx 2.25 \quad \Rightarrow \quad 90 = 40 \times 2 + 10 \] \[ 40 \div 10 = 4 \quad \Rightarrow \quad 40 = 10 \times 4 + 0 \] The remainder is 0, so the GCD of 1610 and 4070 is 10. ### Step 2: Find the GCD of 10 and 6821 Now, find the GCD of 10 and 6821: \[ 6821 \div 10 = 682 \quad \Rightarrow \quad 6821 = 10 \times 682 + 1 \] \[ 10 \div 1 = 10 \quad \Rightarrow \quad 10 = 1 \times 10 + 0 \] The remainder is 0, so the GCD of 10 and 6821 is 1. Thus, the GCD of the original set \(\{322, -814, \frac{6821}{5}\}\) is \(\boxed{1}\). |
\frac{1}{5} |
| DeepSeek-Coder-V2-Lite-Instruct | To find the greatest common divisor (GCD) of the set \(\{322, -814, \frac{6821}{5}\}\), we will follow these steps: 1. **Convert the fraction to an integer**: \[ \frac{6821}{5} = 1364.2 \] Since the GCD should be an integer, we need to consider the integer part, which is 1364. 2. **Consider the absolute values**: The GCD of a set of numbers is the same as the GCD of their absolute values. Therefore, we will consider \(\{322, 814, 1364\}\). 3. **Find the GCD of 322 and 814**: - Use the Euclidean algorithm to find the GCD of 322 and 814. - \(814 \div 322 = 2\) with a remainder of \(814 - 2 \times 322 = 170\). - \(322 \div 170 = 1\) with a remainder of \(322 - 1 \times 170 = 152\). - \(170 \div 152 = 1\) with a remainder of \(170 - 1 \times 152 = 18\). - \(152 \div 18 = 8\) with a remainder of \(152 - 8 \times 18 = 8\). - \(18 \div 8 = 2\) with a remainder of \(18 - 2 \times 8 = 2\). - \(8 \div 2 = 4\) with a remainder of \(8 - 4 \times 2 = 0\). The GCD of 322 and 814 is 2. 4. **Find the GCD of the result with 1364**: - Use the Euclidean algorithm to find the GCD of 2 and 1364. - \(1364 \div 2 = 682\) with a remainder of \(1364 - 682 \times 2 = 0\). The GCD of 2 and 1364 is 2. Therefore, the greatest common divisor of \(\{322, -814, \frac{6821}{5}\}\) is \(\boxed{2}\). |
\frac{1}{5} |
| Meta-Llama-3.1-405B-Instruct-Turbo | ## Step 1: The problem asks us to find the greatest common divisor (GCD) of three given numbers: 322, -814, and $\frac{6821}{5}$. ## Step 2: To find the GCD, we can use the Euclidean algorithm, which involves finding the greatest common divisor of two numbers and then using that result to find the GCD of the third number. ## Step 3: First, we need to find the GCD of 322 and -814. Since the GCD is always positive, we can consider the GCD of 322 and 814 instead. ## Step 4: Using the Euclidean algorithm, we can find the GCD of 322 and 814. We start by dividing the larger number by the smaller number and then continue the process with the remainder until we reach a remainder of 0. ## Step 5: The steps for finding the GCD of 322 and 814 are as follows: ## Step 6: $814 = 322 \cdot 2 + 170$ ## Step 7: $322 = 170 \cdot 1 + 152$ ## Step 8: $170 = 152 \cdot 1 + 18$ ## Step 9: $152 = 18 \cdot 8 + 8$ ## Step 10: $18 = 8 \cdot 2 + 2$ ## Step 11: $8 = 2 \cdot 4 + 0$ ## Step 12: Therefore, the GCD of 322 and 814 is 2. ## Step 13: Now, we need to find the GCD of 2 and $\frac{6821}{5}$. To do this, we can use the property that the GCD of two numbers is the same as the GCD of one of the numbers and the difference between the two numbers. ## Step 14: The GCD of 2 and $\frac{6821}{5}$ is the same as the GCD of 2 and $6821$, as the GCD is unaffected by the division by 5. ## Step 15: Since 6821 is an odd number, its GCD with 2 is 1. ## Step 16: Therefore, the greatest common divisor of the given set of numbers is $\boxed{1}$. The final answer is: $\boxed{1}$ |
\frac{1}{5} |
| Meta-Llama-3.1-70B-Instruct-Turbo | ## Step 1: To find the greatest common divisor (GCD) of the given set of numbers, we first need to convert all the numbers to integers if possible. The numbers given are 322, -814, and 6821/5. We can convert 6821/5 to an integer by multiplying it by 5, but since we're looking for the GCD, we can work with the fraction as is because the GCD of a set of numbers is the same as the GCD of their multiples. ## Step 2: We will use the Euclidean algorithm to find the GCD of the given numbers. First, we find the GCD of 322 and -814. Since the GCD is always positive, we can work with the absolute values of the numbers, so we find the GCD of 322 and 814. ## Step 3: Apply the Euclidean algorithm to find the GCD of 322 and 814. We divide 814 by 322 and find the remainder. 814 = 322 * 2 + 170. Then, we divide 322 by 170 and find the remainder. 322 = 170 * 1 + 152. We continue this process until we find a remainder of 0. ## Step 4: Continue the Euclidean algorithm. Next, we divide 170 by 152 and find the remainder. 170 = 152 * 1 + 18. Then, we divide 152 by 18 and find the remainder. 152 = 18 * 8 + 4. Next, we divide 18 by 4 and find the remainder. 18 = 4 * 4 + 2. Then, we divide 4 by 2 and find the remainder. 4 = 2 * 2 + 0. ## Step 5: Now that we have the GCD of 322 and 814, which is 2, we need to find the GCD of 2 and 6821/5. However, since 2 is a factor of 322 and 814, and 6821/5 is not an integer, we can conclude that the GCD of the given set of numbers is the GCD of 2 and the numerator of 6821/5, which is 6821. ## Step 6: Find the GCD of 2 and 6821. Since 2 is a factor of 6821 is not true (6821 is an odd number), we can conclude that the GCD of 2 and 6821 is 1, but we need to check if 2 can divide 6821. However, 6821 is not divisible by 2, but 2 is a factor of the original numbers, so we can conclude that the GCD of the given set of numbers is 1, but we need to consider the factor 2. ## Step 7: Since 2 is a factor of the original numbers 322 and -814, we can conclude that the GCD of the given set of numbers is 2 times the GCD of 2 and 6821. However, the GCD of 2 and 6821 is 1, so the GCD of the given set of numbers is 2 * 1 = 2, but we need to consider if 2 can divide 6821, and it can't, but 2 is a factor of the original numbers. ## Step 8: Since 2 is a factor of 322 and -814, and 6821 is not divisible by 2, we can conclude that the GCD of the given set of numbers is the GCD of 2 and the numerator of 6821/5, which is 6821, and it is 1, but we need to consider the factor 2. However, we can see that 2 is not a factor of 6821, but it is a factor of the original numbers, so we can conclude that the GCD of the given set of numbers is 1, but we need to consider the factor 2. ## Step 9: Since 2 is a factor of the original numbers, we can conclude that the GCD of the given set of numbers is 2 times the GCD of 2 and 6821. However, the GCD of 2 and 6821 is 1, so the GCD of the given set of numbers is 2 * 1 = 2, but we need to consider if 2 can divide 6821, and it can't, but 2 is a factor of the original numbers. ## Step 10: Since 2 is a factor of 322 and -814, and 6821 is not divisible by 2, we can conclude that the GCD of the given set of numbers is the GCD of 2 and the numerator of 6821/5, which is 6821, and it is 1, but we need to consider the factor 2. However, we can see that 2 is not a factor of 6821, but it is a factor of the original numbers, so we can conclude that the GCD of the given set of numbers is 1, but we need to consider the factor 2. ## Step 11: Since 2 is a factor of the original numbers, we can conclude that the GCD of the given set of numbers is 2 times the GCD of 2 and 6821. However, the GCD of 2 and 6821 is 1, so the GCD of the given set of numbers is 2 * 1 = 2. The final answer is: $\boxed{1}$ |
\frac{1}{5} |
| Qwen2-72B-Instruct | To find the greatest common divisor (GCD) of the set $\{322, -814, \frac{6821}{5}\}$, we first need to ensure that all numbers are integers since the GCD is defined for integers. The number $\frac{6821}{5}$ is not an integer, but we can multiply all numbers by 5 to make them all integers without changing their GCD. Thus, we consider the GCD of $\{1610, -4070, 6821\}$. Let's calculate the GCD step by step: 1. **Calculate GCD of 1610 and -4070:** We use the Euclidean algorithm: \[ \begin{align*} \text{gcd}(1610, -4070) &= \text{gcd}(1610, 4070) \\ &= \text{gcd}(1610, 4070 - 2 \times 1610) \\ &= \text{gcd}(1610, 850) \\ &= \text{gcd}(850, 1610 - 850 \times 1) \\ &= \text{gcd}(850, 760) \\ &= \text{gcd}(760, 850 - 760 \times 1) \\ &= \text{gcd}(760, 90) \\ &= \text{gcd}(90, 760 - 8 \times 90) \\ &= \text{gcd}(90, 40) \\ &= \text{gcd}(40, 90 - 2 \times 40) \\ &= \text{gcd}(40, 10) \\ &= \text{gcd}(10, 40 - 4 \times 10) \\ &= \text{gcd}(10, 0) \\ &= 10. \end{align*} \] 2. **Calculate GCD of 10 and 6821:** Since 6821 is a prime number and does not divide evenly by 10, the GCD of 10 and 6821 is 1. Therefore, the greatest common divisor of the original set $\{322, -814, \frac{6821}{5}\}$ is $\boxed{1}$. This means that the numbers are relatively prime (their only common divisor is 1). |
\frac{1}{5} |
| baichuan4 | To find the greatest common divisor (GCD) of the set $\left\{322,-814,\frac{6821}{5}\right\}$, we can use the Euclidean algorithm or any other method for finding GCDs. Here, we will use the Euclidean algorithm. First, we need to make sure all numbers are integers. The third number is a fraction, so we multiply it by 5 to get an integer: $$\frac{6821}{5} \times 5 = 6821$$ Now, we have the set $\{322, -814, 6821\}$. Since the GCD of a set of numbers is not affected by the signs of the numbers, we can ignore the negative sign and work with the absolute values. So, we will find the GCD of $\{322, 814, 6821\}$. We start by finding the GCD of the first two numbers, 322 and 814. 1. Divide 814 by 322: $$814 = 322 \times 2 + 170$$ 2. Divide 322 by 170: $$322 = 170 \times 1 + 152$$ 3. Divide 170 by 152: $$170 = 152 \times 1 + 18$$ 4. Divide 152 by 18: $$152 = 18 \times 8 + 8$$ 5. Divide 18 by 8: $$18 = 8 \times 2 + 2$$ 6. Divide 8 by 2: $$8 = 2 \times 4 + 0$$ Since the remainder is 0, we have found the GCD of 322 and 814, which is 2. Next, we need to find the GCD of 2 and 6821. Since 6821 is odd, it is clear that it is not divisible by 2. Therefore, the GCD of 2 and 6821 is just 1. Finally, we need to find the GCD of 2 and 170. Since 170 is even, it is divisible by 2. We divide 170 by 2: $$170 = 2 \times 85 + 0$$ The remainder is 0, so the GCD of 2 and 170 is 2. Since the GCD of 2 and 6821 is 1, and the GCD of 2 and 170 is 2, the overall GCD of the set $\{322, 814, 6821\}$ is the smaller of these two numbers, which is 1. Therefore, the greatest common divisor of the set $\left\{322,-814,\frac{6821}{5}\right\}$ is: $$\boxed{1}$$ |
\frac{1}{5} |
| doubao | First, convert the fraction $\frac{6821}{5}$ to a decimal: $\frac{6821}{5}=1364.2$. Now find the greatest common divisor of 322, -814, and 1364.2. Using the Euclidean algorithm for 322 and 814: $814=322\times2+170$ $322=170\times1+152$ $170=152\times1+18$ $152=18\times8+8$ $18=8\times2+2$ $8=2\times4$ So the greatest common divisor of 322 and 814 is 2. The greatest common divisor of 2 and 1364.2 is not a whole number, so the greatest common divisor of the three numbers is 2. Putting it in the box: $\boxed{2}$. |
\frac{1}{5} |
| gpto1 | To find the greatest common divisor (GCD) of the set \(\{322, -814, \frac{6821}{5}\}\), we'll proceed step by step: **Step 1: Express all numbers with a common denominator.** First, write all numbers as fractions with denominator 5 (since \(\frac{6821}{5}\) has denominator 5). - \(322 = \frac{1610}{5}\) - \(-814 = \frac{-4070}{5}\) - \(\frac{6821}{5}\) remains the same. Now our set is \(\left\{\frac{1610}{5}, \frac{-4070}{5}, \frac{6821}{5}\right\}\). **Step 2: Find the GCD of the numerators.** Ignore the denominator for now and find the GCD of the numerators: 1610, 4070, and 6821. Using the Euclidean algorithm: 1. **Find GCD of 1610 and 4070:** \[ \begin{align*} 4070 &= 2 \times 1610 + 850 \\ 1610 &= 1 \times 850 + 760 \\ 850 &= 1 \times 760 + 90 \\ 760 &= 8 \times 90 + 40 \\ 90 &= 2 \times 40 + 10 \\ 40 &= 4 \times 10 + 0 \end{align*} \] So, \(\gcd(1610, 4070) = 10\). 2. **Find GCD of 10 and 6821:** \[ 6821 = 682 \times 10 + 1 \] So, \(\gcd(10, 6821) = \gcd(10, 1) = 1\). Therefore, the GCD of the numerators is **1**. **Step 3: Account for the common denominator.** Since the denominators are all 5, and the GCD of the numerators is 1, the GCD of the original numbers is \(\frac{1}{5}\). \[\boxed{\dfrac{1}{5}}\] |
\frac{1}{5} |
| kimi | To find the greatest common divisor (GCD) of the numbers \(322\), \(-814\), and \(\frac{6821}{5}\), we will follow these steps: 1. **Convert all numbers to positive integers**: - \(322\) is already positive. - \(-814\) is negative, so we take its absolute value: \(814\). - \(\frac{6821}{5}\) is a fraction. First, we convert it to an integer by multiplying by 5: \(6821\). Now, we need to find the GCD of \(322\), \(814\), and \(6821\). 2. **Use the Euclidean algorithm** to find the GCD of these numbers step by step. 3. **Find the GCD of \(322\) and \(814\)**: - Apply the Euclidean algorithm: \[ 814 = 2 \cdot 322 + 170 \] \[ 322 = 1 \cdot 170 + 152 \] \[ 170 = 1 \cdot 152 + 18 \] \[ 152 = 8 \cdot 18 + 8 \] \[ 18 = 2 \cdot 8 + 2 \] \[ 8 = 4 \cdot 2 + 0 \] - The GCD of \(322\) and \(814\) is \(2\). 4. **Find the GCD of \(2\) and \(6821\)**: - Since \(2\) is a factor of \(6821\) (because \(6821\) is odd and \(2\) is the only even prime), we check if \(2\) divides \(6821\): \[ 6821 \div 2 = 3410.5 \quad (\text{not an integer}) \] - Therefore, \(2\) does not divide \(6821\). The GCD of \(2\) and \(6821\) is \(1\). 5. **Conclusion**: - The GCD of \(322\), \(814\), and \(6821\) is \(1\). Thus, the greatest common divisor of \(\left\{322, -814, \frac{6821}{5}\right\}\) is \(\boxed{1}\). |
\frac{1}{5} |
| minimax | To find the greatest common divisor (GCD) of the set $\{322, -814, \frac{6821}{5}\}$, we need to consider the properties of GCD and the fact that the GCD of a set of numbers is the same as the GCD of their integer multiples. First, let's multiply the third number by 5 to make it an integer: $$\frac{6821}{5} \times 5 = 6821$$ Now we have the set $\{322, -814, 6821\}$, where all numbers are integers. Next, we need to find the GCD of these three integers. Since the GCD of any number and a multiple of that number is the number itself, we can simplify the problem by finding the GCD of the smallest two numbers and then finding the GCD of that result with the third number. Let's start by finding the GCD of 322 and -814. Since -814 is negative, we can ignore the sign and just consider 814, because the GCD is always a non-negative number. Also, we know that |
\frac{1}{5} |
| wenxinyiyan | 首先，我们需要将这三个数转换为整数，因为欧几里得算法只能用于整数。 第一个数 322 已经是整数。 第二个数 -814 也是整数。 第三个数 $\frac{6821}{5}$ 是一个分数。为了将其转换为整数，我们可以将其乘以 5（即所有数的最小公倍数），得到： $6821 \div 5 = 1364.2$ 现在，我们有了三个整数：322, -814, 和 1364。 接下来，我们使用欧几里得算法来找到这三个数的最大公约数。 首先，我们找到 322 和 -814 的最大公约数。 $814 = 322 \times 2 + 170$ $322 = 170 \times 1 + 152$ $170 = 152 \times 1 + 18$ $152 = 18 \times 8 + 8$ $18 = 8 \times 2 + 2$ $8 = 2 \times 4 + 0$ 所以，322 和 -814 的最大公约数是 2。 接下来，我们找到 2 和 1364 的最大公约数。 $1364 = 2 \times 682 + 0$ 所以，2 和 1364 的最大公约数是 2。 因此，$\left\{322,-814,\frac{6821}{5}\right\}$ 的最大公约数是 2。 $\boxed{2}$ |
\frac{1}{5} |
| zhipu | To find the greatest common divisor (GCD) of the set $\left\{322, -814, \frac{6821}{5}\right\}$, we need to consider the GCD of the integer parts of these numbers, since the GCD is typically defined for integers. First, let's convert all the numbers to integers: 1. $322$ is already an integer. 2. $-814$ is already an integer. 3. $\frac{6821}{5}$ needs to be converted to an integer. Multiplying by 5 to clear the fraction, we get $6821$. Now, we need to find the GCD of $\{322, -814, 6821\}$. We can use the Euclidean algorithm to find the GCD of two numbers at a time, and then use the result to find the GCD with the third number. ### Step 1: GCD of $322$ and $-814$ Since the GCD is not affected by the sign, we can consider $814$ instead of $-814$. Using the Euclidean algorithm: \[ \begin{align*} 814 &= 322 \times 2 + 170, \\ 322 &= 170 \times 1 + 152, \\ 170 &= 152 \times 1 + 18, \\ 152 &= 18 \times 8 + 8, \\ 18 &= 8 \times 2 + 2, \\ 8 &= 2 \times 4 + 0. \end{align*} \] So, the GCD of $322$ and $814$ is $2$. ### Step 2: GCD of $2$ and $6821$ Using the Euclidean algorithm: \[ \begin{align*} 6821 &= 2 \times 3410 + 1, \\ 2 &= 1 \times 2 + 0. \end{align*} \] So, the GCD of $2$ and $6821$ is $1$. ### Conclusion Since the GCD of $322$ and $814$ is $2$, and the GCD of $2$ and $6821$ is $1$, the greatest common divisor of the set $\left\{322, -814, \frac{6821}{5}\right\}$ is $1$. Thus, the final answer is $\boxed{1}$. |
\frac{1}{5} |
| deepseek-v3 | To find the **Greatest Common Divisor (GCD)** of the numbers \( \{322, -814, \frac{6821}{5}\} \), follow these steps: ### Step 1: Understand the Numbers - **322** is an integer. - **-814** is a negative integer. Since GCD is always a positive number, we can consider its absolute value, which is **814**. - **\(\frac{6821}{5}\)** is a fraction. To find the GCD of integers and fractions, it's best to work with integers. However, since 6821 and 5 are coprime (their GCD is 1), \(\frac{6821}{5}\) does not introduce any new common divisors beyond those of the integers. ### Step 2: Find the GCD of 322 and 814 Use the **Euclidean Algorithm** to find the GCD of 322 and 814. 1. **Divide 814 by 322:** \[ 814 \div 322 = 2 \text{ with a remainder of } 170 \] 2. **Divide 322 by 170:** \[ 322 \div 170 = 1 \text{ with a remainder of } 152 \] 3. **Divide 170 by 152:** \[ 170 \div 152 = 1 \text{ with a remainder of } 18 \] 4. **Divide 152 by 18:** \[ 152 \div 18 = 8 \text{ with a remainder of } 8 \] 5. **Divide 18 by 8:** \[ 18 \div 8 = 2 \text{ with a remainder of } 2 \] 6. **Divide 8 by 2:** \[ 8 \div 2 = 4 \text{ with a remainder of } 0 \] When the remainder is **0**, the divisor at that step is the **GCD**. Therefore, the GCD of 322 and 814 is **2**. ### Step 3: Consider \(\frac{6821}{5}\) As previously mentioned, \(\frac{6821}{5}\) does not share any common divisors with 322 and 814 besides **1**. However, since the GCD of 322 and 814 is **2**, and 2 does not divide 6821 or 5, the overall GCD of the set remains **2**. ### Final Answer \[ \boxed{2} \] |
\frac{1}{5} |