| 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 \(-1403\) and \(751\), 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. Here are the steps: 1. **Divide \(-1403\) by \(751\) and find the remainder:** \[ -1403 \div 751 = -1 \quad \text{(quotient)} \] \[ -1403 - (-1 \times 751) = -1403 + 751 = -652 \] So, the remainder is \(-652\). 2. **Since the remainder is negative, we convert it to a positive remainder by adding \(751\):** \[ -652 + 751 = 99 \] Now, the remainder is \(99\). 3. **Divide \(751\) by \(99\) and find the remainder:** \[ 751 \div 99 \approx 7 \quad \text{(quotient)} \] \[ 751 - (7 \times 99) = 751 - 693 = 58 \] So, the remainder is \(58\). 4. **Divide \(99\) by \(58\) and find the remainder:** \[ 99 \div 58 \approx 1 \quad \text{(quotient)} \] \[ 99 - (1 \times 58) = 99 - 58 = 41 \] So, the remainder is \(41\). 5. **Divide \(58\) by \(41\) and find the remainder:** \[ 58 \div 41 \approx 1 \quad \text{(quotient)} \] \[ 58 - (1 \times 41) = 58 - 41 = 17 \] So, the remainder is \(17\). 6. **Divide \(41\) by \(17\) and find the remainder:** \[ 41 \div 17 \approx 2 \quad \text{(quotient)} \] \[ 41 - (2 \times 17) = 41 - 34 = 7 \] So, the remainder is \(7\). 7. **Divide \(17\) by \(7\) and find the remainder:** \[ 17 \div 7 \approx 2 \quad \text{(quotient)} \] \[ 17 - (2 \times 7) = 17 - 14 = 3 \] So, the remainder is \(3\). 8. **Divide \(7\) by \(3\) and find the remainder:** \[ 7 \div 3 \approx 2 \quad \text{(quotient)} \] \[ 7 - (2 \times 3) = 7 - 6 = 1 \] So, the remainder is \(1\). 9. **Divide \(3\) by \(1\) and find the remainder:** \[ 3 \div 1 = 3 \quad \text{(quotient)} \] \[ 3 - (3 \times 1) = 3 - 3 = 0 \] So, the remainder is \(0\). Since the remainder is now \(0\), the last non-zero remainder is \(1\). Therefore, the greatest common divisor of \(-1403\) and \(751\) is \(\boxed{1}\). |
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| deepseek-r1 | To find the greatest common divisor (GCD) of \(-1403\) and \(751\), we start by taking their absolute values, resulting in \(1403\) and \(751\). We use the Euclidean algorithm: 1. \(1403 \div 751 = 1\) with remainder \(1403 - 751 \times 1 = 652\) 2. \(751 \div 652 = 1\) with remainder \(751 - 652 \times 1 = 99\) 3. \(652 \div 99 = 6\) with remainder \(652 - 99 \times 6 = 58\) 4. \(99 \div 58 = 1\) with remainder \(99 - 58 \times 1 = 41\) 5. \(58 \div 41 = 1\) with remainder \(58 - 41 \times 1 = 17\) 6. \(41 \div 17 = 2\) with remainder \(41 - 17 \times 2 = 7\) 7. \(17 \div 7 = 2\) with remainder \(17 - 7 \times 2 = 3\) 8. \(7 \div 3 = 2\) with remainder \(7 - 3 \times 2 = 1\) 9. \(3 \div 1 = 3\) with remainder \(3 - 1 \times 3 = 0\) The last non-zero remainder is \(1\), so the GCD of \(1403\) and \(751\) is \(1\). Factoring verification confirms \(1403 = 23 \times 61\) and \(751\) is prime, with no common factors other than \(1\). Thus, the greatest common divisor of \(\{-1403, 751\}\) is \(\boxed{1}\). |
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| gpto1 | To find the greatest common divisor (GCD) of \(-1403\) and \(751\), we'll use the Euclidean Algorithm. Since the GCD is always a positive number and unaffected by the sign, we can work with the absolute values: \[ \text{GCD}(1403, 751) \] Now, apply the Euclidean Algorithm: 1. Divide \(1403\) by \(751\) and find the remainder: \[ 1403 = 751 \times 1 + 652 \] 2. Now, take \(751\) and divide it by the remainder \(652\): \[ 751 = 652 \times 1 + 99 \] 3. Continue this process: \[ \begin{align*} 652 &= 99 \times 6 + 58 \\ 99 &= 58 \times 1 + 41 \\ 58 &= 41 \times 1 + 17 \\ 41 &= 17 \times 2 + 7 \\ 17 &= 7 \times 2 + 3 \\ 7 &= 3 \times 2 + 1 \\ 3 &= 1 \times 3 + 0 \end{align*} \] 4. The last non-zero remainder is \(1\), which means: \[ \text{GCD}(1403, 751) = 1 \] Therefore, the greatest common divisor of \(-1403\) and \(751\) is \(\boxed{1}\). |
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| gpt-4o-2024-08-06 | To find the greatest common divisor (GCD) of \(-1403\) and \(751\), 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 need to find the GCD of \(1403\) and \(751\). 1. Divide \(1403\) by \(751\) and find the remainder: \[ 1403 \div 751 \approx 1.868 \quad \Rightarrow \quad \text{quotient} = 1 \] \[ \text{remainder} = 1403 - 751 \times 1 = 1403 - 751 = 652 \] 2. Now, apply the Euclidean algorithm to \(751\) and \(652\): \[ 751 \div 652 \approx 1.151 \quad \Rightarrow \quad \text{quotient} = 1 \] \[ \text{remainder} = 751 - 652 \times 1 = 751 - 652 = 99 \] 3. Next, apply the Euclidean algorithm to \(652\) and \(99\): \[ 652 \div 99 \approx 6.586 \quad \Rightarrow \quad \text{quotient} = 6 \] \[ \text{remainder} = 652 - 99 \times 6 = 652 - 594 = 58 \] 4. Continue with \(99\) and \(58\): \[ 99 \div 58 \approx 1.707 \quad \Rightarrow \quad \text{quotient} = 1 \] \[ \text{remainder} = 99 - 58 \times 1 = 99 - 58 = 41 \] 5. Continue with \(58\) and \(41\): \[ 58 \div 41 \approx 1.414 \quad \Rightarrow \quad \text{quotient} = 1 \] \[ \text{remainder} = 58 - 41 \times 1 = 58 - 41 = 17 \] 6. Continue with \(41\) and \(17\): \[ 41 \div 17 \approx 2.411 \quad \Rightarrow \quad \text{quotient} = 2 \] \[ \text{remainder} = 41 - 17 \times 2 = 41 - 34 = 7 \] 7. Continue with \(17\) and \(7\): \[ 17 \div 7 \approx 2.428 \quad \Rightarrow \quad \text{quotient} = 2 \] \[ \text{remainder} = 17 - 7 \times 2 = 17 - 14 = 3 \] 8. Continue with \(7\) and \(3\): \[ 7 \div 3 \approx 2.333 \quad \Rightarrow \quad \text{quotient} = 2 \] \[ \text{remainder} = 7 - 3 \times 2 = 7 - 6 = 1 \] 9. Finally, continue with \(3\) and \(1\): \[ 3 \div 1 = 3 \quad \Rightarrow \quad \text{quotient} = 3 \] \[ \text{remainder} = 3 - 1 \times 3 = 3 - 3 = 0 \] The last non-zero remainder is \(1\). Therefore, the greatest common divisor of \(-1403\) and \(751\) is \(\boxed{1}\). |
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| DeepSeek-Coder-V2-Lite-Instruct | To find the greatest common divisor (GCD) of \{-1403, 751\}, we can use the Euclidean algorithm, which is a well-known method for finding the GCD of two integers. The Euclidean algorithm is based on the principle that the GCD of two numbers also divides their difference. Here are the steps: 1. **Apply the Euclidean algorithm**: - Start with the two numbers: \( a = 1403 \) and \( b = 751 \). - Since both numbers are positive, we can use the absolute values: \( a = 1403 \) and \( b = 751 \). 2. **Compute the remainder**: - Compute \( 1403 \mod 751 \): \[ 1403 \div 751 = 1 \quad \text{remainder} \quad 1403 - 751 \times 1 = 652 \] - So, \( 1403 \mod 751 = 652 \). 3. **Repeat the process**: - Now, \( a = 751 \) and \( b = 652 \). - Compute \( 751 \mod 652 \): \[ 751 \div 652 = 1 \quad \text{remainder} \quad 751 - 652 \times 1 = 99 \] - So, \( 751 \mod 652 = 99 \). 4. **Continue the process**: - Now, \( a = 652 \) and \( b = 99 \). - Compute \( 652 \mod 99 \): \[ 652 \div 99 = 6 \quad \text{remainder} \quad 652 - 99 \times 6 = 68 \] - So, \( 652 \mod 99 = 68 \). 5. **Continue the process**: - Now, \( a = 99 \) and \( b = 68 \). - Compute \( 99 \mod 68 \): \[ 99 \div 68 = 1 \quad \text{remainder} \quad 99 - 68 \times 1 = 31 \] - So, \( 99 \mod 68 = 31 \). 6. **Continue the process**: - Now, \( a = 68 \) and \( b = 31 \). - Compute \( 68 \mod 31 \): \[ 68 \div 31 = 2 \quad \text{remainder} \quad 68 - 31 \times 2 = 6 \] - So, \( 68 \mod 31 = 6 \). 7. **Continue the process**: - Now, \( a = 31 \) and \( b = 6 \). - Compute \( 31 \mod 6 \): \[ 31 \div 6 = 5 \quad \text{remainder} \quad 31 - 6 \times 5 = 1 \] - So, \( 31 \mod 6 = 1 \). 8. **Continue the process**: - Now, \( a = 6 \) and \( b = 1 \). - Compute \( 6 \mod 1 \): \[ 6 \div 1 = 6 \quad \text{remainder} \quad 6 - 1 \times 6 = 0 \] - So, \( 6 \mod 1 = 0 \). Since the remainder is now 0, the last non-zero remainder is the GCD. Therefore, the greatest common divisor of \{-1403, 751\} is \( \boxed{1} \). |
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| Meta-Llama-3.1-405B-Instruct-Turbo | ## Step 1: To find the greatest common divisor (GCD) of -1403 and 751, we can use the Euclidean algorithm, which is a systematic way of finding the GCD of two numbers. ## Step 2: The Euclidean algorithm involves repeatedly applying the division algorithm, swapping the remainder with the divisor, until the remainder is zero. The last non-zero remainder is the GCD. ## Step 3: We start by taking the absolute values of the given numbers since GCD is defined for positive integers. So, we have |−1403| = 1403 and |751| = 751. ## Step 4: Applying the Euclidean algorithm, we first divide 1403 by 751 to get a quotient and a remainder. Then, we use the remainder as the divisor and the previous divisor as the dividend, and repeat the process until we get a remainder of zero. ## Step 5: The steps of the Euclidean algorithm are as follows: ## Step 6: 1403 = 751 * 1 + 652 ## Step 7: 751 = 652 * 1 + 99 ## Step 8: 652 = 99 * 6 + 58 ## Step 9: 99 = 58 * 1 + 41 ## Step 10: 58 = 41 * 1 + 17 ## Step 11: 41 = 17 * 2 + 7 ## Step 12: 17 = 7 * 2 + 3 ## Step 13: 7 = 3 * 2 + 1 ## Step 14: 3 = 1 * 3 + 0 ## Step 15: The last non-zero remainder is 1, so the greatest common divisor of -1403 and 751 is 1. The final answer is: $\boxed{1}$ |
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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 -1403 and 751, we first apply the Euclidean Algorithm. The algorithm involves dividing the larger number by the smaller number and then replacing the larger number with the remainder of the division. ## Step 2: Perform the first division Divide -1403 by 751 to get the quotient and remainder. -1403 divided by 751 gives a quotient of -1 and a remainder of -652. ## Step 3: Continue the Euclidean Algorithm Now, we divide 751 by -652. This gives a quotient of -1 and a remainder of 99. ## Step 4: Continue the Euclidean Algorithm Next, we divide -652 by 99. This gives a quotient of -6 and a remainder of -58. ## Step 5: Continue the Euclidean Algorithm Then, we divide 99 by -58. This gives a quotient of -1 and a remainder of 41. ## Step 6: Continue the Euclidean Algorithm We divide -58 by 41. This gives a quotient of -1 and a remainder of -17. ## Step 7: Continue the Euclidean Algorithm Next, we divide 41 by -17. This gives a quotient of -2 and a remainder of 7. ## Step 8: Continue the Euclidean Algorithm Then, we divide -17 by 7. This gives a quotient of -2 and a remainder of -3. ## Step 9: Continue the Euclidean Algorithm We divide 7 by -3. This gives a quotient of -2 and a remainder of 1. ## Step 10: Continue the Euclidean Algorithm Next, we divide -3 by 1. This gives a quotient of -3 and a remainder of 0. ## Step 11: Determine the GCD Since the remainder is now 0, the divisor in this step is the GCD. Therefore, the greatest common divisor of -1403 and 751 is 1. The final answer is: $\boxed{1}$ |
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| Qwen2-72B-Instruct | To find the greatest common divisor (GCD) of \(-1403\) and \(751\), we can use the Euclidean algorithm. However, since the GCD is not affected by the sign of the numbers, we can consider the absolute values of these numbers for simplicity. Let's start with the absolute values: \(1403\) and \(751\). Step 1: Divide the larger number by the smaller number and find the remainder. \[1403 = 751 \times 1 + 652\] Step 2: Replace the larger number with the smaller number and the smaller number with the remainder from Step 1. \[751 = 652 \times 1 + 99\] Step 3: Repeat the process. \[652 = 99 \times 6 + 78\] \[99 = 78 \times 1 + 21\] \[78 = 21 \times 3 + 15\] \[21 = 15 \times 1 + 6\] \[15 = 6 \times 2 + 3\] \[6 = 3 \times 2 + 0\] When we reach a remainder of \(0\), the divisor at that step is the GCD. Therefore, the GCD of \(-1403\) and \(751\) is \(3\). So, the answer is \(\boxed{3}\). |
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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. We repeatedly apply this principle until we reach a common divisor. Let's find the GCD of -1403 and 751 using the Euclidean algorithm: 1. Start with the larger absolute value, |-1403| = 1403, and subtract the smaller number, 751, from it: 1403 - 751 = 652 2. Now, take the smaller number, 751, and subtract the new smaller number, 652, from it: 751 - 652 = 99 3. Next, subtract the smaller number, 99, from 652: 652 - 99 = 553 4. Continue subtracting the smaller number, 99, from 553: 553 - 99 = 454 5. Again, subtract 99 from 454: 454 - 99 = 355 6. Subtract 99 from 355: 355 - 99 = 256 7. Subtract 99 from 256: 256 - 99 = 157 8. Subtract 99 from 157: 157 - 99 = 58 9. Subtract 58 from 99: 99 - 58 = 41 10. Subtract 41 from 58: 58 - 41 = 17 11. Finally, subtract 17 from 41: 41 - 17 = 24 12. Subtract 17 from 24: 24 - 17 = 7 13. Subtract 7 from 17: 17 - 7 = 10 14. Subtract 7 from 10: 10 - 7 = 3 15. Subtract 3 from 7: 7 - 3 = 4 16. Subtract 3 from 4: 4 - 3 = 1 At this point, we have reached the number 1, which is the smallest positive integer. This means that the GCD of -1403 and 751 is 1, because the numbers are relatively prime. Therefore, the greatest common divisor of {-1403, 751} is $\boxed{1}$. |
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| doubao | Using the Euclidean algorithm: $1403 = 751 \times 1 + 652$ $751 = 652 \times 1 + 99$ $652 = 99 \times 6 + 58$ $99 = 58 \times 1 + 41$ $58 = 41 \times 1 + 17$ $41 = 17 \times 2 + 7$ $17 = 7 \times 2 + 3$ $7 = 3 \times 2 + 1$ $3 = 1 \times 3$ So the greatest common divisor is 1. $\boxed{1}$. |
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| gpto1 | To find the greatest common divisor (GCD) of \(-1403\) and \(751\), we'll use the Euclidean Algorithm. Since the GCD is always a positive number and unaffected by the sign, we can work with the absolute values: \[ \text{GCD}(1403, 751) \] Now, apply the Euclidean Algorithm: 1. Divide \(1403\) by \(751\) and find the remainder: \[ 1403 = 751 \times 1 + 652 \] 2. Now, take \(751\) and divide it by the remainder \(652\): \[ 751 = 652 \times 1 + 99 \] 3. Continue this process: \[ \begin{align*} 652 &= 99 \times 6 + 58 \\ 99 &= 58 \times 1 + 41 \\ 58 &= 41 \times 1 + 17 \\ 41 &= 17 \times 2 + 7 \\ 17 &= 7 \times 2 + 3 \\ 7 &= 3 \times 2 + 1 \\ 3 &= 1 \times 3 + 0 \end{align*} \] 4. The last non-zero remainder is \(1\), which means: \[ \text{GCD}(1403, 751) = 1 \] Therefore, the greatest common divisor of \(-1403\) and \(751\) is \(\boxed{1}\). |
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| kimi | To find the greatest common divisor (GCD) of \(-1403\) and \(751\), we can use the Euclidean algorithm. The Euclidean algorithm involves repeated division and taking remainders until we reach a remainder of zero. The last non-zero remainder is the GCD. Here are the steps: 1. **Identify the absolute values**: Since the GCD of two numbers is the same as the GCD of their absolute values, we consider \(1403\) and \(751\). 2. **Apply the Euclidean algorithm**: - Divide \(1403\) by \(751\) and find the remainder. \[ 1403 \div 751 \approx 1.866 \quad \text{(integer part is 1)} \] \[ 1403 - 751 \times 1 = 1403 - 751 = 652 \] So, the remainder is \(652\). 3. **Repeat the process**: - Now, divide \(751\) by \(652\) and find the remainder. \[ 751 \div 652 \approx 1.153 \quad \text{(integer part is 1)} \] \[ 751 - 652 \times 1 = 751 - 652 = 99 \] So, the remainder is \(99\). 4. **Continue the process**: - Next, divide \(652\) by \(99\) and find the remainder. \[ 652 \div 99 \approx 6.585 \quad \text{(integer part is 6)} \] \[ 652 - 99 \times 6 = 652 - 594 = 58 \] So, the remainder is \(58\). 5. **Repeat again**: - Now, divide \(99\) by \(58\) and find the remainder. \[ 99 \div 58 \approx 1.707 \quad \text{(integer part is 1)} \] \[ 99 - 58 \times 1 = 99 - 58 = 41 \] So, the remainder is \(41\). 6. **Continue**: - Next, divide \(58\) by \(41\) and find the remainder. \[ 58 \div 41 \approx 1.415 \quad \text{(integer part is 1)} \] \[ 58 - 41 \times 1 = 58 - 41 = 17 \] So, the remainder is \(17\). 7. **Final step**: - Finally, divide \(41\) by \(17\) and find the remainder. \[ 41 \div 17 \approx 2.412 \quad \text{(integer part is 2)} \] \[ 41 - 17 \times 2 = 41 - 34 = 7 \] So, the remainder is \(7\). 8. **Continue until the remainder is zero**: - Divide \(17\) by \(7\) and find the remainder. \[ 17 \div 7 \approx 2.428 \quad \text{(integer part is 2)} \] \[ 17 - 7 \times 2 = 17 - 14 = 3 \] So, the remainder is \(3\). 9. **Final step**: - Divide \(7\) by \(3\) and find the remainder. \[ 7 \div 3 \approx 2.333 \quad \text{(integer part is 2)} \] \[ 7 - 3 \times 2 = 7 - 6 = 1 \] So, the remainder is \(1\). 10. **Final step**: - Divide \(3\) by \(1\) and find the remainder. \[ 3 \div 1 = 3 \quad \text{(integer part is 3)} \] \[ 3 - 1 \times 3 = 3 - 3 = 0 \] So, the remainder is \(0\). Since the last non-zero remainder is \(1\), the greatest common divisor of \(-1403\) and \(751\) is \(\boxed{1}\). |
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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 -1403 and 751. 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 1403 and 751. Now, we repeatedly apply the Euclidean algorithm: 1. Divide the larger number by the smaller number and take the remainder: $$ 1403 = 751 \cdot 1 + 652 $$ 2. Replace the larger number with the smaller number and the smaller number with the remainder from the previous step: $$ 751 = 652 \cdot 1 + 99 $$ 3. Continue this process: $$ 652 = 99 \cdot 6 + 58 $$ $$ 99 = 58 \cdot 1 + 41 $$ $$ 58 = 41 \cdot 1 + 17 $$ $$ 41 = 17 \cdot 2 + 7 $ |
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| wenxinyiyan | 首先，我们观察两个数，$-1403$ 和 $751$。 为了找到这两个数的最大公约数，我们可以使用欧几里得算法。欧几里得算法是一个古老且有效的方法，用于计算两个整数的最大公约数。 按照欧几里得算法，我们首先计算两个数的绝对值，然后执行以下步骤： 1. 用较大的数除以较小的数，取余数。 2. 然后用较小的数除以这个余数，再次取余数。 3. 重复这个过程，直到余数为0。 在这个过程中，最后一个非零余数就是这两个数的最大公约数。 对于 $-1403$ 和 $751$，我们首先计算它们的绝对值，得到 $1403$ 和 $751$。 现在，我们开始执行欧几里得算法： 1. $1403 \div 751 = 1$ 余 $652$ 2. $751 \div 652 = 1$ 余 $99$ 3. $652 \div 99 = 6$ 余 $58$ 4. $99 \div 58 = 1$ 余 $41$ 5. $58 \div 41 = 1$ 余 $17$ 6. $41 \div 17 = 2$ 余 $7$ 7. $17 \div 7 = 2$ 余 $3$ 8. $7 \div 3 = 2$ 余 $1$ 9. $3 \div 1 = 3$ 余 $0$ 在这个过程中，最后一个非零余数是 $1$。 因此，$-1403$ 和 $751$ 的最大公约数是 $1$。 所以，最终答案是 $\boxed{1}$。 |
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| zhipu | We use the Euclidean Algorithm to find the greatest common divisor of $-1403$ and $751$. Since the GCD is not affected by the signs of the numbers, we can work with positive values for convenience. Thus, we consider the GCD of $1403$ and $751$. \[ \text{gcd}(1403, 751) = \text{gcd}(1403 - 751, 751) \] \[ = \text{gcd}(652, 751) \] \[ = \text{gcd}(652, 751 - 652) \] \[ = \text{gcd}(652, 99) \] \[ = \text{gcd}(652 - 6 \cdot 99, 99) \] \[ = \text{gcd}(58, 99) \] \[ = \text{gcd}(58, 99 - 58) \] \[ = \text{gcd}(58, 41) \] \[ = \text{gcd}(58 - 41, 41) \] \[ = \text{gcd}(17, 41) \] Since $41$ is a prime number and does not divide $17$, the greatest common divisor is $1$. Thus, the greatest common divisor of $\{-1403, 751\}$ is $\boxed{1}$. |
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| deepseek-v3 | Error code: 400 - {'error': {'code': 'RequestTimeOut', 'param': None, 'message': 'Request timed out, please try again later.', 'type': 'RequestTimeOut'}, 'id': 'chatcmpl-bbc463d1-d9a6-9b15-906d-96cb9eea4622', 'request_id': 'bbc463d1-d9a6-9b15-906d-96cb9eea4622'} | 1 |