| 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 numbers \(-894\) and \(1322\), we will use the Euclidean algorithm. The Euclidean algorithm involves repeated division and taking remainders until the remainder is zero. The last non-zero remainder is the GCD. Let's start with the given numbers: 1. Compute \(1322 \mod -894\): \[ 1322 \div -894 \approx -1.478 \quad \text{(take the integer part, which is -1)} \] \[ 1322 - (-894 \times -1) = 1322 - 894 = 428 \] So, \(1322 \mod -894 = 428\). 2. Now, compute \(894 \mod 428\): \[ 894 \div 428 \approx 2.089 \quad \text{(take the integer part, which is 2)} \] \[ 894 - (428 \times 2) = 894 - 856 = 38 \] So, \(894 \mod 428 = 38\). 3. Next, compute \(428 \mod 38\): \[ 428 \div 38 \approx 11.263 \quad \text{(take the integer part, which is 11)} \] \[ 428 - (38 \times 11) = 428 - 418 = 10 \] So, \(428 \mod 38 = 10\). 4. Then, compute \(38 \mod 10\): \[ 38 \div 10 \approx 3.8 \quad \text{(take the integer part, which is 3)} \] \[ 38 - (10 \times 3) = 38 - 30 = 8 \] So, \(38 \mod 10 = 8\). 5. Next, compute \(10 \mod 8\): \[ 10 \div 8 \approx 1.25 \quad \text{(take the integer part, which is 1)} \] \[ 10 - (8 \times 1) = 10 - 8 = 2 \] So, \(10 \mod 8 = 2\). 6. Finally, compute \(8 \mod 2\): \[ 8 \div 2 = 4 \quad \text{(exact division)} \] \[ 8 - (2 \times 4) = 8 - 8 = 0 \] So, \(8 \mod 2 = 0\). Since the remainder is now zero, the last non-zero remainder is the GCD. Therefore, the greatest common divisor of \(-894\) and \(1322\) is: \[ \boxed{2} \] |
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| deepseek-r1 | To find the greatest common divisor (GCD) of \(\{-894, 1322\}\), we first take the absolute value of -894, which gives us 894. We then use the Euclidean algorithm to find the GCD of 894 and 1322. 1. Divide 1322 by 894: \[ 1322 = 894 \times 1 + 428 \] The remainder is 428. So, \(\gcd(1322, 894) = \gcd(894, 428)\). 2. Divide 894 by 428: \[ 894 = 428 \times 2 + 38 \] The remainder is 38. So, \(\gcd(894, 428) = \gcd(428, 38)\). 3. Divide 428 by 38: \[ 428 = 38 \times 11 + 10 \] The remainder is 10. So, \(\gcd(428, 38) = \gcd(38, 10)\). 4. Divide 38 by 10: \[ 38 = 10 \times 3 + 8 \] The remainder is 8. So, \(\gcd(38, 10) = \gcd(10, 8)\). 5. Divide 10 by 8: \[ 10 = 8 \times 1 + 2 \] The remainder is 2. So, \(\gcd(10, 8) = \gcd(8, 2)\). 6. Divide 8 by 2: \[ 8 = 2 \times 4 + 0 \] The remainder is 0. So, \(\gcd(8, 2) = 2\). The last non-zero remainder is 2, which means the GCD of 894 and 1322 is 2. Therefore, the GCD of \(\{-894, 1322\}\) is also \(\boxed{2}\). |
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| gpto1 | To find the greatest common divisor (GCD) of \(-894\) and \(1322\), we can use the Euclidean algorithm, which works the same for negative numbers by considering their absolute values. Let's find \(\gcd(894, 1322)\): 1. **Step 1:** \(1322 ÷ 894 = 1\) remainder \(1322 - 894 × 1 = 428\) \[ 1322 = 1 × 894 + 428 \] 2. **Step 2:** \(894 ÷ 428 = 2\) remainder \(894 - 428 × 2 = 38\) \[ 894 = 2 × 428 + 38 \] 3. **Step 3:** \(428 ÷ 38 = 11\) remainder \(428 - 38 × 11 = 10\) \[ 428 = 11 × 38 + 10 \] 4. **Step 4:** \(38 ÷ 10 = 3\) remainder \(38 - 10 × 3 = 8\) \[ 38 = 3 × 10 + 8 \] 5. **Step 5:** \(10 ÷ 8 = 1\) remainder \(10 - 8 × 1 = 2\) \[ 10 = 1 × 8 + 2 \] 6. **Step 6:** \(8 ÷ 2 = 4\) remainder \(8 - 2 × 4 = 0\) \[ 8 = 4 × 2 + 0 \] The last non-zero remainder is \(2\), so \(\gcd(894, 1322) = 2\). Therefore, the greatest common divisor of \(-894\) and \(1322\) is: \(\boxed{2}\) |
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| gpt-4o-2024-08-06 | To find the greatest common divisor (GCD) of \(-894\) and \(1322\), we can use the Euclidean algorithm. The Euclidean algorithm involves repeated division and taking remainders until the remainder is zero. The last non-zero remainder is the GCD. First, we take the absolute values of the numbers since the GCD is always non-negative. So, we work with \(894\) and \(1322\). 1. Divide \(1322\) by \(894\) and find the remainder: \[ 1322 \div 894 \approx 1.479 \quad \Rightarrow \quad \text{quotient} = 1 \] \[ \text{remainder} = 1322 - 1 \times 894 = 1322 - 894 = 428 \] 2. Now, apply the Euclidean algorithm to \(894\) and \(428\): \[ 894 \div 428 \approx 2.089 \quad \Rightarrow \quad \text{quotient} = 2 \] \[ \text{remainder} = 894 - 2 \times 428 = 894 - 856 = 38 \] 3. Next, apply the Euclidean algorithm to \(428\) and \(38\): \[ 428 \div 38 \approx 11.263 \quad \Rightarrow \quad \text{quotient} = 11 \] \[ \text{remainder} = 428 - 11 \times 38 = 428 - 418 = 10 \] 4. Continue with \(38\) and \(10\): \[ 38 \div 10 \approx 3.8 \quad \Rightarrow \quad \text{quotient} = 3 \] \[ \text{remainder} = 38 - 3 \times 10 = 38 - 30 = 8 \] 5. Continue with \(10\) and \(8\): \[ 10 \div 8 \approx 1.25 \quad \Rightarrow \quad \text{quotient} = 1 \] \[ \text{remainder} = 10 - 1 \times 8 = 10 - 8 = 2 \] 6. Finally, apply the Euclidean algorithm to \(8\) and \(2\): \[ 8 \div 2 = 4 \quad \Rightarrow \quad \text{quotient} = 4 \] \[ \text{remainder} = 8 - 4 \times 2 = 8 - 8 = 0 \] The remainder is now zero, and the last non-zero remainder is \(2\). Therefore, the greatest common divisor of \(-894\) and \(1322\) is \(\boxed{2}\). |
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| DeepSeek-Coder-V2-Lite-Instruct | To find the greatest common divisor (GCD) of \{-894, 1322\}, we can use the Euclidean algorithm, which involves repeated division. Here are the steps: 1. **Apply the Euclidean algorithm:** - Start with the two numbers: \( a = 1322 \) and \( b = 894 \). - Divide \( a \) by \( b \) and find the remainder: \( 1322 \div 894 = 1 \) with a remainder of \( 1322 - 894 \times 1 = 428 \). - Now, set \( a = 894 \) and \( b = 428 \). - Divide \( a \) by \( b \) and find the remainder: \( 894 \div 428 = 2 \) with a remainder of \( 894 - 428 \times 2 = 38 \). - Now, set \( a = 428 \) and \( b = 38 \). - Divide \( a \) by \( b \) and find the remainder: \( 428 \div 38 = 11 \) with a remainder of \( 428 - 38 \times 11 = 10 \). - Now, set \( a = 38 \) and \( b = 10 \). - Divide \( a \) by \( b \) and find the remainder: \( 38 \div 10 = 3 \) with a remainder of \( 38 - 10 \times 3 = 8 \). - Now, set \( a = 10 \) and \( b = 8 \). - Divide \( a \) by \( b \) and find the remainder: \( 10 \div 8 = 1 \) with a remainder of \( 10 - 8 \times 1 = 2 \). - Now, set \( a = 8 \) and \( b = 2 \). - Divide \( a \) by \( b \) and find the remainder: \( 8 \div 2 = 4 \) with a remainder of \( 8 - 2 \times 4 = 0 \). 2. **Conclusion:** - Since the remainder is now 0, the last non-zero remainder is the GCD. - Therefore, the greatest common divisor of \{-894, 1322\} is \( 2 \). Thus, the final answer is $\boxed{2}$. |
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| Meta-Llama-3.1-405B-Instruct-Turbo | ## Step 1: To find the greatest common divisor (GCD) of -894 and 1322, we first need to understand what a GCD is. The GCD of two numbers is the largest positive integer that divides both numbers without leaving a remainder. ## Step 2: Since we're dealing with a negative number, we should note that the GCD is always positive. Therefore, we will find the GCD of the absolute values of the given numbers, which are 894 and 1322. ## Step 3: To calculate the GCD, we can use the Euclidean algorithm, which involves repeatedly applying the division algorithm to find the remainder and then using the remainder as the divisor in the next step. ## Step 4: We start by dividing the larger number by the smaller number and then continue the process with the smaller number and the remainder until we obtain a remainder of 0. The last non-zero remainder will be the GCD. ## Step 5: Applying the Euclidean algorithm: ## Step 6: We start with the larger number, 1322, as the dividend and the smaller number, 894, as the divisor. ## Step 7: 1322 = 894 * 1 + 428 ## Step 8: Next, we use the previous divisor, 894, as the dividend and the remainder, 428, as the divisor. ## Step 9: 894 = 428 * 2 + 38 ## Step 10: Continuing the process, we use the previous divisor, 428, as the dividend and the remainder, 38, as the divisor. ## Step 11: 428 = 38 * 11 + 10 ## Step 12: Next, we use the previous divisor, 38, as the dividend and the remainder, 10, as the divisor. ## Step 13: 38 = 10 * 3 + 8 ## Step 14: Continuing, we use the previous divisor, 10, as the dividend and the remainder, 8, as the divisor. ## Step 15: 10 = 8 * 1 + 2 ## Step 16: Next, we use the previous divisor, 8, as the dividend and the remainder, 2, as the divisor. ## Step 17: 8 = 2 * 4 + 0 ## Step 18: Since we have obtained a remainder of 0, the last non-zero remainder, which is 2, is the GCD of 894 and 1322. ## Step 19: Therefore, the greatest common divisor of -894 and 1322 is 2. The final answer is: $\boxed{2}$ |
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| Meta-Llama-3.1-70B-Instruct-Turbo | ## Step 1: Apply the Euclidean Algorithm To find the greatest common divisor (GCD) of -894 and 1322, we will use the Euclidean Algorithm. This algorithm involves repeatedly applying the division algorithm to find the remainder, then using the divisor and remainder as the new dividend and divisor until the remainder is 0. ## Step 2: Perform the First Iteration We start by dividing the larger number by the smaller one: 1322 divided by 894. However, since we are looking for the GCD of the absolute values of these numbers, we will consider the positive values. The first step involves dividing 1322 by 894. ## Step 3: Calculate the First Remainder Performing the division: 1322 ÷ 894 = 1 remainder 428. The remainder is 428. ## Step 4: Perform the Second Iteration Next, we divide the divisor by the remainder from the previous step: 894 divided by 428. ## Step 5: Calculate the Second Remainder Performing the division: 894 ÷ 428 = 2 remainder 38. The remainder is 38. ## Step 6: Perform the Third Iteration Now, we divide the previous divisor by the remainder: 428 divided by 38. ## Step 7: Calculate the Third Remainder Performing the division: 428 ÷ 38 = 11 remainder 10. The remainder is 10. ## Step 8: Perform the Fourth Iteration Next, we divide the previous divisor by the remainder: 38 divided by 10. ## Step 9: Calculate the Fourth Remainder Performing the division: 38 ÷ 10 = 3 remainder 8. The remainder is 8. ## Step 10: Perform the Fifth Iteration Now, we divide the previous divisor by the remainder: 10 divided by 8. ## Step 11: Calculate the Fifth Remainder Performing the division: 10 ÷ 8 = 1 remainder 2. The remainder is 2. ## Step 12: Perform the Sixth Iteration Next, we divide the previous divisor by the remainder: 8 divided by 2. ## Step 13: Calculate the Sixth Remainder Performing the division: 8 ÷ 2 = 4 remainder 0. The remainder is 0. ## Step 14: Identify the GCD Since the last non-zero remainder is 2, the greatest common divisor of 894 and 1322 is 2. The final answer is: $\boxed{2}$ |
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| Qwen2-72B-Instruct | To find the greatest common divisor (GCD) of \(-894\) and \(1322\), we can ignore the sign because the GCD is always positive. So, we're looking for the GCD of \(894\) and \(1322\). We can use the Euclidean algorithm to find the GCD: 1. Divide the larger number by the smaller number and find the remainder. \[1322 = 894 \times 1 + 428\] 2. Replace the larger number with the smaller number and the smaller number with the remainder from step 1, then repeat the process. \[894 = 428 \times 2 + 38\] 3. Continue this process until the remainder is \(0\). \[428 = 38 \times 11 + 0\] When the remainder is \(0\), the divisor at that step is the GCD. So, the GCD of \(894\) and \(1322\) is \(38\). Therefore, the greatest common divisor of \(-894\) and \(1322\) is \(\boxed{38}\). |
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| baichuan4 | To find the greatest common divisor (GCD) of two numbers, we can use the Euclidean algorithm. This algorithm is based on the principle that the GCD of two numbers also divides their difference. Let's apply the Euclidean algorithm to find the GCD of -894 and 1322: 1. Start with the larger number (absolute value), which is 1322, and subtract the smaller number (absolute value), which is 894: $$ 1322 - (-894) = 1322 + 894 = 2216 $$ 2. Now, take the absolute value of the smaller number, 894, and divide it into the result from the previous step: $$ 2216 \div 894 = 2 \text{ with a remainder of } 428 $$ 3. Next, take the absolute value of the remainder, 428, and divide it into the absolute value of the previous divisor, 894: $$ 894 \div 428 = 2 \text{ with a remainder of } 38 $$ 4. Finally, take the absolute value of the remainder, 38, and divide it into the absolute value of the previous divisor, 428: $$ 428 \div 38 = 11 \text{ with a remainder of } 10 $$ 5. Since 38 does not divide evenly into 10, we stop here. The GCD is the last non-zero divisor, which is 38. Therefore, the greatest common divisor of -894 and 1322 is 38. $$\boxed{38}$$ |
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| doubao | Using the Euclidean algorithm: $1322 = 1 \times 894 + 428$ $894 = 2 \times 428 + 138$ $428 = 3 \times 138 + 14$ $138 = 9 \times 14 + 12$ $14 = 1 \times 12 + 2$ $12 = 6 \times 2$ So the greatest common divisor of $-894$ and $1322$ is $2$. $\boxed{2}$. |
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| gpto1 | To find the greatest common divisor (GCD) of \(-894\) and \(1322\), we can use the Euclidean algorithm, which works the same for negative numbers by considering their absolute values. Let's find \(\gcd(894, 1322)\): 1. **Step 1:** \(1322 ÷ 894 = 1\) remainder \(1322 - 894 × 1 = 428\) \[ 1322 = 1 × 894 + 428 \] 2. **Step 2:** \(894 ÷ 428 = 2\) remainder \(894 - 428 × 2 = 38\) \[ 894 = 2 × 428 + 38 \] 3. **Step 3:** \(428 ÷ 38 = 11\) remainder \(428 - 38 × 11 = 10\) \[ 428 = 11 × 38 + 10 \] 4. **Step 4:** \(38 ÷ 10 = 3\) remainder \(38 - 10 × 3 = 8\) \[ 38 = 3 × 10 + 8 \] 5. **Step 5:** \(10 ÷ 8 = 1\) remainder \(10 - 8 × 1 = 2\) \[ 10 = 1 × 8 + 2 \] 6. **Step 6:** \(8 ÷ 2 = 4\) remainder \(8 - 2 × 4 = 0\) \[ 8 = 4 × 2 + 0 \] The last non-zero remainder is \(2\), so \(\gcd(894, 1322) = 2\). Therefore, the greatest common divisor of \(-894\) and \(1322\) is: \(\boxed{2}\) |
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| kimi | To find the greatest common divisor (GCD) of \(-894\) and \(1322\), we can follow these steps: 1. **Ignore the signs**: The GCD of two numbers is the same regardless of their signs. Therefore, we can consider the absolute values of the numbers: \[ \text{GCD}(-894, 1322) = \text{GCD}(894, 1322) \] 2. **Use the Euclidean algorithm**: The Euclidean algorithm is an efficient method for finding the GCD of two numbers. It is based on the principle that \(\text{GCD}(a, b) = \text{GCD}(b, a \mod b)\). 3. **Apply the Euclidean algorithm**: - First, divide \(1322\) by \(894\) and find the remainder: \[ 1322 \div 894 = 1 \quad \text{with a remainder of} \quad 1322 - 894 \times 1 = 428 \] So, \(\text{GCD}(894, 1322) = \text{GCD}(894, 428)\). - Next, divide \(894\) by \(428\) and find the remainder: \[ 894 \div 428 = 2 \quad \text{with a remainder of} \quad 894 - 428 \times 2 = 38 \] So, \(\text{GCD}(894, 428) = \text{GCD}(428, 38)\). - Then, divide \(428\) by \(38\) and find the remainder: \[ 428 \div 38 = 11 \quad \text{with a remainder of} \quad 428 - 38 \times 11 = 10 \] So, \(\text{GCD}(428, 38) = \text{GCD}(38, 10)\). - Next, divide \(38\) by \(10\) and find the remainder: \[ 38 \div 10 = 3 \quad \text{with a remainder of} \quad 38 - 10 \times 3 = 8 \] So, \(\text{GCD}(38, 10) = \text{GCD}(10, 8)\). - Then, divide \(10\) by \(8\) and find the remainder: \[ 10 \div 8 = 1 \quad \text{with a remainder of} \quad 10 - 8 \times 1 = 2 \] So, \(\text{GCD}(10, 8) = \text{GCD}(8, 2)\). - Finally, divide \(8\) by \(2\) and find the remainder: \[ 8 \div 2 = 4 \quad \text{with a remainder of} \quad 8 - 2 \times 4 = 0 \] So, \(\text{GCD}(8, 2) = 2\). 4. **Conclusion**: Since the remainder is now \(0\), the GCD is the last non-zero remainder, which is \(2\). Therefore, the greatest common divisor of \(-894\) and \(1322\) is \(\boxed{2}\). |
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| minimax | To find the greatest common divisor (GCD) of two numbers, we can use the Euclidean algorithm. Let's apply it to the numbers -894 and 1322. First, we take the absolute values of the numbers since the GCD is always positive, and the algorithm works the same way for positive numbers. The absolute values are 894 and 1322. Now, we repeatedly apply the Euclidean algorithm: 1. Divide the larger number by the smaller number and find the remainder: $$ 1322 \div 894 = 1 \text{ with a remainder of } 428 $$ 2. Replace the larger number with the smaller number and the smaller number with the remainder from the previous step: Now we consider the pair (894, 428). 3. Repeat the process: $$ 894 \div 428 = 2 \text{ with a remainder of } 38 $$ 4. Continue the process: $$ 428 \div 38 = 11 \text{ with a remainder of } 10 $$ $$ 38 \div 10 = |
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| wenxinyiyan | 为了找到-894和1322的最大公约数，我们可以使用欧几里得算法。 首先，我们计算两个数的绝对值，因为最大公约数与数的正负无关。所以，我们考虑894和1322。 接下来，应用欧几里得算法： 1322 = 894 × 1 + 428 894 = 428 × 2 + 38 428 = 38 × 11 + 10 38 = 10 × 3 + 8 10 = 8 × 1 + 2 8 = 2 × 4 + 0 由此，我们可以看到最后的余数为0，所以最大公约数是2。 故答案为：$\boxed{2}$。 |
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| zhipu | We use the Euclidean Algorithm to find the greatest common divisor of $-894$ and $1322$. \begin{align*} \gcd(-894, 1322) &= \gcd(-894, 1322 - (-894)) \\ &= \gcd(-894, 1322 + 894) \\ &= \gcd(-894, 2216) \\ &= \gcd(-894 + 4 \cdot 2216, 2216) \\ &= \gcd(6430, 2216) \\ &= \gcd(6430 - 2 \cdot 2216, 2216) \\ &= \gcd(202, 2216) \\ &= \gcd(202, 2216 - 10 \cdot 202) \\ &= \gcd(202, 216) \\ &= \gcd(202 - 216, 216) \\ &= \gcd(-14, 216) \\ &= \gcd(14, 216) \\ &= \gcd(14, 216 - 15 \cdot 14) \\ &= \gcd(14, 6) \\ &= \gcd(14 - 2 \cdot 6, 6) \\ &= \gcd(2, 6) \\ &= \boxed{2}. \end{align*} |
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| deepseek-v3 | To find the **greatest common divisor (GCD)** of the numbers \(-894\) and \(1322\), we can use the **Euclidean Algorithm**. Here's a step-by-step solution: ### Step 1: Apply the Euclidean Algorithm The Euclidean Algorithm states that the GCD of two numbers also divides their difference. We can ignore the negative sign of \(-894\) since the GCD is always a positive integer. So, we start with: \[ \gcd(-894, 1322) = \gcd(894, 1322) \] ### Step 2: Divide and Find Remainders We proceed by dividing the larger number by the smaller number and replacing the larger number with the remainder. 1. **First Division:** \[ 1322 \div 894 = 1 \quad \text{with a remainder of} \quad 1322 - 894 \times 1 = 428 \] So, \[ \gcd(894, 1322) = \gcd(894, 428) \] 2. **Second Division:** \[ 894 \div 428 = 2 \quad \text{with a remainder of} \quad 894 - 428 \times 2 = 38 \] So, \[ \gcd(894, 428) = \gcd(428, 38) \] 3. **Third Division:** \[ 428 \div 38 = 11 \quad \text{with a remainder of} \quad 428 - 38 \times 11 = 10 \] So, \[ \gcd(428, 38) = \gcd(38, 10) \] 4. **Fourth Division:** \[ 38 \div 10 = 3 \quad \text{with a remainder of} \quad 38 - 10 \times 3 = 8 \] So, \[ \gcd(38, 10) = \gcd(10, 8) \] 5. **Fifth Division:** \[ 10 \div 8 = 1 \quad \text{with a remainder of} \quad 10 - 8 \times 1 = 2 \] So, \[ \gcd(10, 8) = \gcd(8, 2) \] 6. **Sixth Division:** \[ 8 \div 2 = 4 \quad \text{with a remainder of} \quad 8 - 2 \times 4 = 0 \] Since the remainder is now \(0\), the process stops. ### Step 3: Determine the GCD The last non-zero remainder is \(2\). Therefore, the greatest common divisor of \(-894\) and \(1322\) is: \[ \boxed{2} \] |
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